Dynamics of meromorphic maps with small topological degree III: geometric currents and ergodic theory
aa r X i v : . [ m a t h . D S ] S e p DYNAMICS OF MEROMORPHIC MAPS WITH SMALL TOPOLOGICALDEGREE III: GEOMETRIC CURRENTS AND ERGODIC THEORY
JEFFREY DILLER, ROMAIN DUJARDIN, AND VINCENT GUEDJ
Abstract.
We continue our study of the dynamics of mappings with small topological de-gree on projective complex surfaces. Previously, under mild hypotheses, we have constructedan ergodic “equilibrium” measure for each such mapping. Here we study the dynamical prop-erties of this measure in detail: we give optimal bounds for its Lyapunov exponents, provethat it has maximal entropy, and show that it has product structure in the natural extension.Under a natural further assumption, we show that saddle points are equidistributed towardsthis measure. This generalizes results that were known in the invertible case and adds tothe small number of situations in which a natural invariant measure for a non-invertibledynamical system is well-understood.
Introduction
In this article we continue our investigation, begun in [DDG1, DDG2], of dynamics oncomplex surfaces for rational transformations with small topological degree. Our previouswork culminated in the construction of a canonical mixing invariant measure for a very broadclass of such mappings. We intend now to study in detail the nature of this measure. As we willshow, the measure meets conjectural expectations concerning, among other things, Lyapunovexponents, entropy, product structure in the natural extension, and equidistribution of saddleorbits.Before entering into the details of our results, let us recall our setting. Let X be a complexprojective surface (always compact and connected), and f : X → X be a rational mapping.Our main requirement is that f has small topological degree :(1) λ ( f ) < λ ( f ) . Here the topological (or second dynamical) degree λ ( f ) is the number of preimages of ageneric point, whereas the first dynamical degree λ ( f ) := lim (cid:13)(cid:13) ( f n ) ∗ | H , ( X ) (cid:13)(cid:13) /n measuresthe asymptotic volume growth of preimages of curves under iteration of f . We refer the readerto [DDG1] for a more precise discussion of dynamical degrees. In particular it was observedthere that the existence of maps with small topological degree imposes some restrictions on theambient surface: either X is rational (in particular, projective), or X has Kodaira dimensionzero.Let us recall that the ergodic theory of mappings with large topological degree ( λ > λ )has been extensively studied, and that results analogous to our Theorems B and C are true in Date : November 7, 2018.2000
Mathematics Subject Classification.
Key words and phrases. dynamics of meromorphic mappings, laminar and woven currents, entropy, naturalextension.J.D. acknowldeges the National Science Foundation for its support through grant DMS 06-53678.R.D. was partially supported by ANR through project BERKO. this context [BrDu, DS, G1]. In dimension 1, all rational maps have large topological degree,and in this setting these results are due to [Ly, FLM]. We note also that in the birationalcase λ = 1, the main results of this paper are obtained in [BLS1, BLS2] (for polynomialautomorphisms of C ) in [Ca] (for automorphisms of projective surfaces) and in [Du4] (forgeneral birational maps). Hence the focus here is on noninvertible mappings which, as thereader will see, present substantial additional difficulties.We will work under two additional assumptions, which we now introduce. Good birational model.
We need to assume that the linear actions ( f n ) ∗ induced by f n on cohomology are compatible with the dynamics, i.e.( H1) ( f n ) ∗ = ( f ∗ ) n , for all n ∈ N . This condition, often called “algebraic stability” in the literature, was first considered byFornaess and Sibony [FS]. There is some evidence that for any mapping (
X, f ) with smalltopological degree, there should exist a birationally conjugate mapping ( ˜ X, ˜ f ) that satisfies(H1). Birational conjugacy does not affect dynamical degrees, so in this case we simply replacethe given system ( X, f ) with the “good birational model” ( ˜ X, ˜ f ).We observed in [DDG1] that the minimal model for X is a good birational model when X has Kodaira dimension zero. For rational X , there is a fairly explicit blowing up proceedure[DF] that produces a good model when λ = 1. More recently, Favre and Jonsson [FJ] haveprove that each polynomial mapping of C with small topological degree admits a good modelon passing to an iterate.Under assumption (H1), in [DDG1], we have constructed and studied canonical invariantcurrents T + and T − . These are defined by T + = lim c + λ n ( f n ) ∗ ω and T − = lim c − λ n ( f n ) ∗ ω where ω is a fixed K¨ahler form on X and c ± are normalizing constants chosen so that incohomology { T + } · { T − } = { ω } · { T − } = { ω } · { T + } = 1. A fact of central importance to usis that these currents have additional geometric structures: T + is laminar, while T − is woven(see below § Finite energy.
Let I + denote the indeterminacy set of f , i.e. the collection of those pointsthat f “blows up” to curves; and let I − denote the analogous set of points which are imagesof curves under f . The invariant current T + (resp. T − ) typically has positive Lelong numberat each point of the extended indeterminacy set I + ∞ = S n ≥ f − n I + (resp. I −∞ = S n ≥ f n I − ).Condition (H1) is equivalent to asking that the sets I + ∞ and I −∞ be disjoint.In order to give meaning to and study the wedge product T + ∧ T − , it is desirable to havemore quantitative control on how fast these sets approach one another. This is how our nexthypothesis should be understood:( H2) f has finite dynamical energy.We refer the reader to [DDG2] for a precise definition of finite energy and its relationshipwith recurrence properties of indeterminacy points. In that article we proved the followingtheorem. Theorem A ([DDG2]) . Let f be a meromorphic map with small topological degree on aprojective surface, satisfying hypotheses (H1) and (H2). Then the wedge product µ := T + ∧ T − is a well-defined probability invariant measure that is f -invariant and mixing. Furthermore EOMETRIC CURRENTS AND ERGODIC THEORY 3 the wedge product is described by the geometric intersection of the laminar/woven structuresof T + and T − . The notion of “geometric intersection” will be described at length in § ⋄ We can now state the main results of this article. Let us emphasize that they rely onthe hypotheses (H1) and (H2) only through the conclusions of Theorem A. Taking theseconclusions as a starting point, one can read the proofs given here independently of [DDG1,DDG2].
Theorem B.
Let X be a complex projective surface and f : X → X be a rational map withsmall topological degree. Assume that f satisfies the conclusions of Theorem A. Then thecanonical invariant measure µ = T + ∧ T − has the following properties:i. For µ -a.e. p there exists a nonzero tangent vector e s at p , such that (2) lim sup n →∞ n log | df n ( e s ( p )) | ≤ − log( λ /λ )2 . ii. Likewise, for µ -a.e. p there exist a tangent vector e u at p , and a set of integers N ′ ⊂ N of density 1 such that (3) lim inf N ′ ∋ n →∞ n log | df n ( e u ( p )) | ≥ log λ . iii. µ has entropy log λ ; thus it has maximal entropy and h top ( f ) = log λ .iv. The natural extension of µ has local product structure. In particular it follows from iv. and the work of Ornstein and Weiss [OW] (see also Briend[Br] for useful remarks on the adaptation to the noninvertible case) that the natural extensionof µ has the Bernoulli property, hence µ is mixing to all orders and has the K property. Aprecise definition of local product structure will be given below in §
8. This is the analogue ofthe balanced property of the maximal measure in the large topological degree case.Let us stress that we do not assume that log dist( · , I + ∪ C f ) is µ integrable ( C f denotesthe critical set). This condition is usually imposed to guarantee the existence of Lyapunovexponents and applicability of the Pesin theory of non-uniformy hyperbolic dynamical sys-tems. However, for mappings with small topological degree, it is known to fail in general (see[DDG2, § χ + ( µ ) ≥ χ − ( µ ) are well defined, then (i) and (ii) implythat χ + ( µ ) ≥
12 log λ ( f ) > > −
12 log λ ( f ) /λ ( f ) ≥ χ − ( µ ) , hence the measure µ is hyperbolic. These bounds are optimal and were conjectured in [G3].In order to go further and relate µ to the distribution of saddle periodic points, we usePesin theory and must therefore invoke the above integrability hypothesis. Theorem C.
Under the assumptions of Theorem B, assume further that ( H3) p log dist( p, I + ∪ C f ) ∈ L ( µ ) , where C f is the critical set. JEFFREY DILLER, ROMAIN DUJARDIN, AND VINCENT GUEDJ
Then, for every n there exists a set P n ⊂ Supp( µ ) of saddle periodic points of period n ,with P n ∼ λ n , and such that λ n X q ∈P n δ q −→ µ. Let
Per n be the set of all isolated periodic points of f of period n . If furthermore - f has no curves of periodic points, - or X = P or P × P ,then n ∼ λ n , so that asymptotically nearly all periodic points are saddles. This theorem was proved for birational maps by the second author in [Du4] (though thepossibility of a curve of periodic points was overlooked there). It would be interesting toprove a similar result without using Pesin Theory (i.e. without assumption (H3)).It would also be interesting to know when saddle points might lie outside Supp( µ ). Onecan easily create isolated saddle points by blowing up an attracting fixed point with unequaleigenvalues. We then get an infinitely near saddle point in the direction corresponding tothe larger multiplier, whose unstable manifold is contained in the exceptional divisor of theblow-up. We do not know any example of a saddle point outside Supp( µ ) whose stable andunstable manifolds are both Zariski dense.While the results in this paper parallel those in [Du4], new and more elaborate argumentsare needed for non-invertible maps. In particular, we are led to work in the natural extension(e.g. for establishing iii. and iv. of Theorem B ), in a situation where there is no symmetrybetween the preimages along µ (see the examples in § · , I + ∪ C f ), so that it is delicate to workwith. Therefore it is reasonable to restrict the uniqueness problem to measures satisfying thisassumption. However, even with this restriction, and even with the additional assumptionthat f is birational, the problem remains unsolved. ⋄ EOMETRIC CURRENTS AND ERGODIC THEORY 5
We now discuss applicability of our assumptions. For mappings on Kodaira zero surfaces,we have seen that we can always assume that (H1) is satisfied. It is actually not very hardto prove that (H2) and (H3) are also always true (see [DDG1, Proposition 4.8] and [DDG2,Proposition 4.5]). Thus Theorems B and C yield the following:
Corollary D.
Let X be a complex projective surface of Kodaira dimension zero. Let f : X → X be a rational transformation with small topological degree. Then the conclusions ofTheorems B and C hold for f . When X is rational our results apply notably to the case where f is the rational extension ofa polynomial mapping of C with small topological degree. As noted above, Favre and Jonssonhave proven that a slightly weaker variation of (H1) holds for f in a suitable compactification X of C . As we show in [DDG1, § f k , the currents T + and T − are actually invariant by f . Since (H2) depends on f only through the currents T + and T − , and (H3) holds for f as soon as it holds for an iterate, we conclude from [DDG2] that (H2) and (H3), hence the conclusions of Theorem A, hold for f . Altogether this impliesthe following corollary. Corollary E.
Let f : C → C be a polynomial mapping with small topological degree. Thenthe conclusions of Theorems B and C apply to f . Let us emphasize that the small topological degree assumption is needed here: the readerwill find in [G3] an easy example of a polynomial endomorphism f : C → C with λ ( f ) = λ ( f ) > h top ( f ) = 0.Regarding more general rational mappings on rational surfaces, we noted above there aregrounds to suspect that (H1) holds generally, after suitable birational conjugation. Amongmaps satisfying (H1), there is no known example of a map that violates the energy condi-tion (H2). We offer some evidence in [DDG2] that (H2) might only fail in very degeneratesituations. On the other hand, we give examples [DDG2, § ⋄ The structure of this paper is as follows. § § § § § § Acknowledgments.
We are grateful to the referee for a very careful reading and helpfulcomments. R.D. also thanks J.-P. Thouvenot for useful conversations.1.
Preliminaries on geometric currents
We begin by collecting some general facts about geometric, that is laminar and woven,currents. We often use the single word “current” as a shorthand for “positive closed (1,1)current on a complex surface.” [DDG2] was accepted before [DDG1], so statements about polynomial maps in that article are still givenin terms of iterates. JEFFREY DILLER, ROMAIN DUJARDIN, AND VINCENT GUEDJ
Laminations and laminar currents.
Recall that a lamination by Riemann surfaces is a topological space such that every point admits a neighborhood U α homeomorphic (by φ α )to a product of the form D × τ α (with coordinates ( z, t )), where τ α is some locally compact set, D is the unit disk, and such that the transition maps φ α ◦ φ − β are of the form ( h ( z, t ) , h ( t )),with h holomorphic in the disk direction z . By definition a plaque is a subset of the form φ − α ( D × { t } ), and a flow box is a subset of the form φ − α ( D × K ), with K a compact set.A leaf is a minimal connected set L with the property that every plaque intersecting L iscontained in L . An invariant transverse measure is given by a collection of measures on thetransverse sets τ α , compatible with the transition maps φ α ◦ φ − β . The survey by Ghys [Gh] isa good reference for these notions. We always assume that the space is separable, so that is iscovered by countably many flow boxes. In this paper we will consider “abstract” laminationsas well as laminations embedded in complex surfaces. In the latter case we require of coursethat the complex structure along the plaques is compatible with the ambient one.Two flow boxes embedded in a manifold are said to be compatible if the correspondingplaques intersect along open sets. Notice that disjoint flow boxes are compatible by definition.A weak lamination is a countable union of compatible flow boxes. It makes sense to speak ofleaves and invariant transverse measures on a weak lamination. Being primarily interested inmeasure-theoretic properties, we needn’t distinguish between ordinary and weak laminationsin this paper.Let us also recall that a (1 ,
1) current T is uniformly laminar if it is given by integrationover an embedded lamination endowed with an invariant transverse measure. That is, therestriction T | φ − ( τ × D ) to a single flow box can be expressed Z τ [ φ − ( { t } × D )] dµ τ ( t ) , where µ τ is the measure induced by the transverse measure on τ .The current T is laminar if it is an integral over a measurable family of compatible holo-morphic disks. Equivalently, for each ε > X ε ⊂ X and a uni-formly laminar current T ε ≤ T in X ε such that the mass (in X ) of the difference satisfies M ( T − T ε ) < ε . It is a key fact that the laminar currents we consider in this paper have someadditional geometric properties. For instance, each has a natural underlying weak lamination,and the lamination carries an invariant transverse measure. We refer the reader to [Du3, Du4]for details about this.Our main purpose in this section is to explore the related notion of woven current andgeneralize some results of [Du3] that we will need afterwards.1.2. Marked woven currents.
Given an open set Q in a Hermitian complex surface, we let Z ( Q, C ) denote the set of (codimension 1) analytic chains with volume bounded by C . Weendow Z ( Q, C ) with the topology of currents. Since there is a dense sequence of test forms,this topology is metrizable. Most often we identify a chain and its support, which is a closedanalytic subset of Q . We denote by ∆( Q, C ) ⊂ Z ( Q, C ) the closure of the set of analyticdisks in Z ( Q, C ).By definition, a uniformly woven current in Q is an integral of integration currents overchains in Z ( Q, C ), for some C [Di]. A woven current is an increasing limit of sums of uniformlywoven currents. EOMETRIC CURRENTS AND ERGODIC THEORY 7
A given laminar current can be expressed as an integral of disks in an essentially uniquefashion (only reparameterizations up to a set of zero measure are possible [BLS1, Lemma6.5]). For woven (even uniformly) currents, this is not true. For example, ω = idz ∧ d ¯ z + idw ∧ d ¯ w = 12 id ( z + w ) ∧ d ( z + w ) + 12 id ( z − w ) ∧ d ( z − w ) . That is, the standard K¨ahler form in C may be written as a sum of uniformly laminarcurrents in two very different ways.We say that a uniformly woven current is marked if it is presented as an integral of disks.More specifically, a marking for a uniformly woven current T in some open set Q is a positiveBorel measure m ( T ) on Z ( Q, C ) for some C such that T = Z Z ( Q,C ) [ D ] dm ( T )( D ) . Abusing terminology, we will also refer to m ( T ) as the transverse measure associated to T . We call the support of m ( T ) the web supporting T . The woven currents considered inthis paper have the additional property of being strongly approximable (see § Q, C ).We define strong convergence for marked uniformly woven currents as follows: T n stronglyconverges to T if the markings m ( T n ) are supported in Z ( Q, C ) for a fixed C and convergeweakly to m ( T ). We leave the reader to check that this implies the usual convergence ofcurrents.1.3. Markings for strongly approximable woven currents.
The woven structure forthe invariant current T − has several additional properties, which play a crucial role in thepaper. We say that a current T is a strongly approximable woven current if it is obtained asa limit of divisors [ C n ] /d n whose geometric genus is O ( d n ) (here by definition the geometricgenus of a chain is the sum of the genera of its components). See [DDG1, §
3] for a proof that T − is of this type.Its woven structure can be constructed as follows. We fix two generically transverse linearprojections π i : X → P , and subdivisions by squares of the projection bases. For each square S ⊂ P , we discard from the approximating sequence [ C n ] /d n all connected components π − i ( S ) ∩ C n over S which are not graphs of area ≤ /
2. The geometric assumption on C n implies that the corresponding loss of mass is small. Taken together, the two projectionsdivide the ambient manifold into a collection Q of “cubes” of size r and these partition theremaining well-behaved part of [ C n ] /d n into a collection of uniformly woven currents T Q,n whose sum T Q ,n closely approximates [ C n ] /d n . The disks constituting T Q ,n will be referredto as “good components”.More specifically we define a cube to be a subset of the form Q := π − ( S ) ∩ π − ( S ),where the S i are squares. Near a point where π and π are regular and the squares S i aresmall enough, Q is actually biholomorphic to an affine cube. The woven current T Q,n , or moreprecisely its marking m ( T Q,n ), is defined by assigning mass 1 /d n to each ‘good’ componentof C n ∩ Q . Then we have the mass estimate M (cid:0) T Q ,n − [ C n ] /d n (cid:1) = O ( r ), independent of n ,where r is the size of the cubes.There is a subtle point here. The number of disks constituting C n ∩ Q might be much largerthan d n . Thus the masses of the measures m ( T Q,n ) might not be bounded above uniformlyin n . However, by Lelong’s theorem [Lel] the disks intersecting a smaller subcube Q ′ have JEFFREY DILLER, ROMAIN DUJARDIN, AND VINCENT GUEDJ volume bounded from below, so there are no more than ≤ c ( Q ′ , Q ) d n of these. Restricting themarking to these disks gives rise to a new current, which we continue to denote by T Q,n , thatcoincides with the old one on Q ′ . The mass of m ( T Q,n ) is now locally uniformly bounded in Z ( Q, / m ( T Q ) denote the limitingmeasure and T Q the corresponding current. Let T Q be the sum of these currents, where Q ranges over the subdivision Q . We then have that M (cid:0) T Q − T (cid:1) = O ( r ).Finally, we consider a increasing sequence of subdivisions by cubes Q i of size r i →
0, andby the previous procedure we obtain an increasing sequence of currents T Q i , converging to T by the previous estimate. Notice that in the case of T + the construction is the same, exceptthat at each step the disks constituting the T Q ,n are disjoint so that the T Q are uniformlylaminar.Since the approximating disks are restrictions of graphs over each projection, the markingof T Q has the following additional virtues. Lemma 1.1.
The multiplicity with which disks in
Supp m ( T Q,n ) converge to chains belongingto Supp m ( T Q ) is always equal to one. Similarly, if a chain D ∈ Supp( m ( T Q )) is a Hausdorfflimit of other chains D n ∈ Supp( m ( T Q )) , then the multiplicity of convergence is equal to one.Thus there is no folding; i.e. in either cases the corresponding tangent spaces also converge.In particular the chains in Supp( m ( T Q )) are non-singular. On the other hand, the Hausdorff limit of a sequence of disks might not be itself a diskbut rather a chain with several components. As an example, consider the family of parabolas w = z + c , c ∈ C , each restricted to the (open) unit bidisk. For | c | < | c | →
1, it becomes disconnected. Thefollowing observation will be useful to us in several places.
Lemma 1.2.
For generic subdivisons by squares of the projection bases of π and π , we havethat m ( T Q ) -almost every chain is a disk (i.e. has only one component).Proof. Recall that each chain in the support of m ( T Q,n ) is obtained by intersecting a graphover some square in the base of e.g. π with the preimage π − ( S ) of a square in the base of π .So if a sequence of disks has a disconnected limit, then the limit must be a piece of a graphtangent to some fiber of π . However, a graph over π that is not outright contained in afiber of π will be tangent to at most countably many fibers of π . Since we have uncountablymany choices for the subdivision by squares associated to π , the result follows from standardmeasure theory arguments. (cid:3) Geometric intersection.
Let Q ⊂ C be an open subset, and D , D ′ be two holomor-phic chains in Q . We define [ D ] ˙ ∧ [ D ′ ] as the sum of point masses, counting multiplicities, atisolated intersections of D and D ′ . Given more generally two marked uniformly woven cur-rents T and T , with associated measures m and m , we define the geometric intersection as T ˙ ∧ T = Z [ D ] ˙ ∧ [ D ] ( dm ⊗ dm ) ( D , D ) . In general it is necessary to take multiplicities into account because of the possibility ofpersistently non transverse intersection of chains. Notice that the definition depends not onlyon the currents but also on the markings.The following basic proposition says that under reasonable assumptions the wedge productof uniformly woven currents is geometric.
EOMETRIC CURRENTS AND ERGODIC THEORY 9
Proposition 1.3. If T and T are as above and if T ∈ L ( T ) , then T ∧ T = T ˙ ∧ T . See [DDG2, Prop. 2.6] for a proof. As a corollary, if T ∈ L ( T ), the geometric wedgeproduct T ˙ ∧ T is independent of the markings.Since woven currents are less well-behaved than laminar ones, we do not try as in the lam-inar case [Du4] to give an intrinsic definition of the geometric intersection of woven currents.Instead we focus only on the particular situation that arises in this paper: we have stronglyapproximable currents T + (laminar) and T − (woven) in X , whose geometric structures areobtained by extracting good components of approximating curves, as explained in the previ-ous subsection. Let Q i be the increasing sequence of subdivisions by cubes constructed as in § T ±Q i be the corresponding currents. Let us write T + Q ˙ ∧ T −Q for the sum of T + Q ˙ ∧ T − Q over Q ∈ Q .Under the finite energy hypothesis (H2) from the introduction, we have shown in [DDG2]that the wedge product T + ∧ T − is a well-defined probability measure. Though the currents T ±Q i depend on the choice of generic linear projections, of generic subdivisions for the projec-tion bases and of convergent subsequences extracted from T ±Q i ,n , the central result of [DDG2, § T + Q i ˙ ∧ T −Q i increase to T + ∧ T − regardless. We summarize the factsthat the limiting measure has the correct mass and is independent of choices by saying that the intersection of T + and T − is geometric .A slightly delicate point is that in [DDG2], T + ∧ T − is not always defined in the usual L fashion. In particular it is not clear whether T + Q ∈ L ( T − Q ) for Q ∈ Q i . Nevertheless[DDG2, Lemma 2.10] implies that this is true after negligible modification of the markings.So throughout the paper we assume that for every i and every Q ∈ Q i , T + Q ∈ L ( T − Q ) sothat by Proposition 1.3 we can always identify T + Q ∧ T − Q and T + Q ˙ ∧ T − Q .We now prove the useful fact that the geometric product of uniformly woven currentsis lower semicontinuous with respect to the strong topology. It can further be seen that,except for boundary effects, discontinuities can occur only when the chains have commoncomponents. Lemma 1.4.
Let Q ⋐ Q ′ be open sets and T , T be marked, uniformly woven currents in Q with markings supported on ∆( Q ′ , / . Then ( T , T ) Z Q T ˙ ∧ T is lower semicontinuous with respect to the strong topology.Proof. Observe first that for
D, D ′ ∈ ∆( Q ′ , / R Q [ D ] ˙ ∧ [ D ′ ] on Q is lower semicon-tinuous in the strong topology. This is just a fancy way of saying that isolated intersectionsbetween D and D ′ persist under small perturbation.Now for the general case, suppose that T j , T j strongly converge to T , T , and let m j , m j be the corresponding transverse measures. Then Z Q T j ˙ ∧ T j = Z ∆( Q ′ , / (cid:18)Z Q [ D ] ˙ ∧ [ D ′ ] (cid:19) d ( m j ⊗ m j ) . Now by strong convergence, we have that m j ⊗ m j converges to m ⊗ m weakly; and we havejust seen that the inner integral is lower semicontinuous in D, D ′ . Hence the lemma follows Here genericity is understood in the measure-theoretic sense. from a well-known bit of measure theory: if ν j is a sequence of positive Radon measures withuniformly bounded masses, weakly converging to some ν , and if ϕ is any lower semicontinuousfunction, then lim inf j →∞ h ν j , ϕ i ≥ h ν, ϕ i . (cid:3) We will also need the following lemma.
Lemma 1.5. If L is a generic hyperplane section of X , the wedge product T + ∧ [ f k ( L )] iswell defined for all k ∈ N (in the L sense) and gives no mass to points.Proof. Since [ f k ( L )] = f k ∗ [ L ] for any L disjoint from I ( f k ), it suffices by invariance of T + toprove the result for k = 0. Fix a projective embedding X ֒ → P ℓ . Then the hyperplane sectionson X are parametrized by the dual P ℓ ∗ . If dv is Fubini-Study volume on P ℓ ∗ , then the Croftonformula says that α := R P ℓ ∗ [ L ] dv ( L ) is the restriction to X of the Fubini-Study K¨ahler formon P ℓ . Hence T + ∧ [ L ] is well-defined for almost all L and T + ∧ α = R T + ∧ [ L ] dv ( L ).To show that T + ∧ [ L ] does not charge points for generic L , we exploit laminarity . Since α is smooth, we have that T + Q i ∧ α increases to T + ∧ α as i → ∞ . Thus for almost any [ L ], T + Q i ∧ [ L ] increases to T + ∧ [ L ]. In particular, T + Q i ∧ [ L ] is well-defined, and since T + Q i is a sumof uniformly laminar currents, the intersection is geometric. Since T + does not charge curves, T + Q i is diffuse and T + Q i ˙ ∧ [ L ] therefore gives no mass to points. Hence neither does T + ∧ [ L ]. (cid:3) The tautological bundle.
Let T be a marked uniformly woven current in Q , withassociated measure m ( T ) as above. The tautological bundle over Z ( Q, C ) is the (closed) setˇ Z ( Q, C ) = { ( D, p ) , p ∈ Supp( D ) } ⊂ Z ( Q, C ) × Q. Similarly we define the tautological bundle ˇ T over T by restricting to Supp m ( T ). Definingthe tautological bundle is a somewhat artificial way of separating the disks of T , which willnevertheless be quite useful conceptually.In particular, when passing to the tautological bundle, the web supporting T becomesa weak lamination with transverse measure m ( T ). When (as in our situation) there is nofolding, we get a lamination.Let ˇ σ T be the product of the area measure along the disks D with m ( T ). Let ˇ π : ˇ T → Q be the natural projection. Then ˇ π ∗ ˇ σ T is the usual trace measure σ T of T . For σ T a.e. p ∈ Q ,ˇ σ T induces a conditional measure ˇ σ T ( ·| p ) on the fiber ˇ π − ( p ), which records as a measure theset of disks passing through p .1.6. Analytic continuation property.
The dynamical results in [Du4] depend on fine prop-erties of strongly approximable uniformly laminar currents proved in [Du3]. The situationis similar here. Therefore we now state and sketch the proof of an “analytic continuation”property for curves subordinate to strongly approximable woven currents. This is a fairlystraightforward extension of [Du3] (see in particular Remark 3.13 there). Hence our briefaccount closely follows the more detailed presentation in § § C n ] /d n of curves withslowly growing genera converging to our current T , a generic linear projection π : X → P ,and approximations T Q i ,n , T Q i of [ C n ] /d n and T corresponding to a sequence of subdivisions Q i of P into squares Q . Note that in order to better cohere with notation in [Du3], we aredeparting from our overall convention by letting Q denote a square in P rather than a cubein X . Likewise, the notation T Q (resp. T Q , etc.) refers to a current in π − ( Q ) that is a limit This fact does not seem to follow trivially from the fact that T + does not charge curves EOMETRIC CURRENTS AND ERGODIC THEORY 11 of families of good components of C n over Q . We may assume that each fiber of π has unitarea and intersects C n in d n points counting multiplicity.Let W be a web , i.e. an arbitrary union of smooth curves in some open set . For anysquare Q ⊂ P , we define the restriction T Q | W by restricting the marking measure for T Q tothe set of disks contained in W . We then take T Q i | W = P Q ∈Q i T Q | W and define T | W to bethe increasing limit of T Q i | W . Thus T | W depends on the choice of marking. One can check,however, that it does not depend on the sequence of subdivisions Q i .Here is the analytic continuation statement we will need. Theorem 1.6. T | W is a uniformly woven current.Proof (sketch). We assume without loss of generality that the leaves Γ α of W are graphsover a disk U of area less than 1 /
2. Note that this allows us to view a restriction T Q | W as the restriction T Q, W | Q of a marked uniformly woven current T Q, W defined over all of U . Moreover, rather than work with a single sequence of subdivisions, we choose three suchsequences ( Q ji ) i ∈ N , j = 1 , , P .So given any x ∈ P , we can choose squares Q decreasing to x and define the marked uniformlywoven current T x, W to be the increasing limit of the currents T Q, W . Given another uniformlywoven current S = R Γ α ds ( α ) supported on W , we will say that T strongly dominates S over Q (resp. over x ) if the marking for T Q, W (resp. for T x, W ) dominates ds .Let us also recall the notion of “defect”. If Q ⊂ P is a square, then the defect dft( Q, n )is the fraction of those components of C n over Q that are bad. By strong convergence,dft( Q, n ) converges to a limit dft( Q ) as n → ∞ . Slow growth of genera implies that the totalnumber of bad components over all squares in a subdivision is not larger than Cd n . Hencethe total defect P Q ∈Q i dft( Q ) of T Q i is bounded uniformly in i . Then we can define thedefect dft( x ) = lim Q ց x dft( Q ) of T over a point x ∈ P . The total defect bound becomes P x ∈ P dft( x ) ≤ C . In particular, dft( x ) is positive for only countably many x ∈ X .The key fact underlying Theorem 1.6 is (compare [Du3, Prop. 3.10] and also [Duv]) Lemma 1.7.
Suppose that x , x ∈ P are points satisfying dft( x ) = dft( x ) = 0 . Then T x , W = T x , W as marked uniformly woven currents. One proves this lemma by choosing a path γ from x to x along which T has zero defectat every point. Given ε >
0, one can then choose a finite cover of γ by squares Q ∋ x , Q , . . . , Q N ∋ x such that P dft( Q j ) < ε , and such that the mass of T Q , W is within ε ofthat of T x , W . Beginning with T Q , W and Q , one then proceeds from Q to Q and so on,keeping only that part of T Q , W supported on leaves that remain good over the new square.At each step one loses mass proportional to the defect over the squares involved. In the end,one arrives at Q N with a marked uniformly woven current S strongly dominated by each ofthe T Q j , W but with the mass bound M ( T Q , W − S ) < Cε . Letting ε decrease to zero, onethen infers that T x , W strongly dominates T x , W . By symmetry, we have equality.From the lemma, we have a marked uniformly woven current T W = T x, W that is indepen-dent of which zero defect point we use to define it. Since each square contains points with zerodefect, we have that T W strongly dominates T Q, W for every Q . In particular T | W ≤ T W . Onthe other hand, the argument used to prove the lemma gives a slightly different statement: if Q is a rectangle with dft( Q ) < δ , then there is a marked uniformly woven current S stronglydominated by T Q, W such that M ( T W − S ) < Cδ . To prove this, one applies the argument We will only need the statement for webs of smooth curves, but the case of singular W could be of interest. from the previous paragraph to the trivial path from some zero defect point x = x ∈ Q “covered” by two squares Q , Q such that Q = Q and dft( Q ) is arbitrarily small.Now we can choose a finite number of points x , . . . , x N ∈ P such that P x = x j dft( x ) < ε .Since neither T nor T W charge fibers of π , the modified version of the lemma tells that P Q ∈Q ji ,x j / ∈ Q T Q | W , which is a woven current dominated by both T | W and T W , is within mass Cε of T W when i is large. Shrinking ε , we infer T | W = T W as desired. (cid:3) Preliminaries on ergodic theory
We now collect some standard facts from measurable dynamics that will be useful to us.2.1.
The natural extension.
A good reference for this paragraph is the (yet unpublished)book of M. Urbanski and F. Przytycki [PU, Chapter 1].The natural extension of a (non-invertible) measurable dynamical system (
X, µ, f ) is the(unique up to isomorphism) invertible system ( ˆ X, ˆ µ, ˆ f ) semiconjugate to ( X, µ, f ) by a pro-jection ˆ π : ˆ X → X with the universal property that any other semiconjugacy ̟ : Y → X ofan invertible system ( Y, ν, g ) onto (
X, µ, f ) factors through ˆ π .The natural extension ˆ X may be presented concretely as the space of histories , i.e. se-quences ( x n ) n ≤ such that f ( x n ) = x n +1 . Here ˆ π = π is the projection (( x n ) n ≤ ) = x ontothe 0 th factor; ˆ f is the shift map ( x j ) ( x j +1 ); and ˆ µ is the unique ˆ f -invariant measuresuch that ˆ µ ( π − ( A )) = µ ( A ). From this point of view, the factorization of a semiconjugacy η : Y → X by some other invertible system is easy to describe. For each y ∈ Y , we have ̟ ( y ) = ˆ π ◦ η ( y ), where η ( y ) is the sequence ( ̟ ( g n ( y ))) n ≤ .The natural extension preserves entropy: h ˆ µ ( ˆ f ) = h µ ( f ). Also ( ˆ f , ˆ µ ) is ergodic iff ( f, µ ) is.Another charaterization of ˆ µ is the following. It corresponds to the presentation of ( ˆ X, ˆ µ, ˆ f )as the inverse limit of the system of measure preserving maps ( · · · X f → X · · · ). Consider thestandard model of the natural extension, and denote by π − n the projection on the ( − n ) t h factor. Lemma 2.1. If ν is any probability measure on ˆ X such that for every n ≥ , ( π − n ) ∗ ν = µ ,then ν = ˆ µ .Proof. Let C A − n ,...,A = (cid:8) ˆ x ∈ ˆ X, ∀ i ≤ n, x − i ∈ A − i (cid:9) be a cylinder of depth n . One verifieseasily that under the assumption of the lemma, ν ( C A − n ,...,A ) ≤ ˆ µ ( C A − n ,...,A ). From this weinfer that ν ≤ ˆ µ , hence ν = ˆ µ by equality of the masses. (cid:3) Measurable partitions and conditional measures.
We will use the formalism ofmeasurable partitions and conditional measures, so we recall a few facts (see [BLS1] for a shortpresentation, and [Ro, PU] for a more systematic treatment). Recall first that a
Lebesgue space is a probability space isomorphic to the unit interval with Lebesgue measure, plus countablymany atoms. All the spaces we will consider in the paper are Lebesgue. A measurable partition of a Lebesgue space is the partition into the fibers of some measurable function. If ξ is ameasurable partition, a probability measure ν may be disintegrated with respect to ξ , givingrise to a probability measure ν ξ ( x ) on almost every atom of ξ (the conditional measure). Thefunction x R φ ( y ) dν ξ ( x ) ( y ) is measurable and we have the following disintegration formula:for every continuous function φ , Z (cid:18)Z φ ( y ) dν ξ ( x ) ( y ) (cid:19) dν ( x ) = Z φdν. EOMETRIC CURRENTS AND ERGODIC THEORY 13
Conversely, the validity of this formula for all φ characterizes the conditional measures.Given partitions ξ i , we denote by W ξ i the joint partition , i.e. ( W ξ i ) ( x ) = T ( ξ i ( x )).If π is a measurable map (possibly between different spaces) and ξ a measurable partition,we define the (measurable) partition π − ξ by ( π − ξ )( p ) = π − ( ξ ( p )). We have the followingeasy lemma, whose proof is left to the reader. Lemma 2.2.
Let ( ˜
Y , ˜ ν ) , ( Y, ν ) be two probability spaces with a measure preserving map π : ˜ Y → Y . Assume that ξ is a measurable partition of Y , and denote by ˜ ξ the measurablepartition π − ( ξ ) .Then for ˜ ν -a.e. p ∈ ˜ Y , π ∗ (˜ ν ˜ ξ ( p ) ) = ν ξ ( π ( p )) . Outline of proof of Theorem B.
Before embarking to the proof, we give an overview of the main arguments leading toTheorem B. The proof of i. ( §
4) is based on the study of the contraction properties alongdisks subordinate to the laminar current T + . This is delicate but fairly similar to [Du4], andit is achieved in § ii. is in the same spirit but with many more differences. The fact that thecurrent T − is only woven leads to substantial difficulties, the first of which is that there is nonatural web associated to such a current. This has been overcome by De Th´elin in [dT1] whohas given a short argument leading to ii. Nevertheless to compute the entropy and establish local product structure we need a fineranalysis of µ and its natural extension. We therefore take a longer path. For the bounds onboth positive and negative exponents, we use an argument “`a la Lyubich” that is suitableonly for showing contraction. So for the positive exponent, we need to iterate backward. Tomake this possible, we first carefully select ( §
5) a set of distinguished histories which has fullmeasure in the natural extension, and exhibits exponential contraction along disks. Then ( § T − to construct a dynamical system ( ˇ f , ˇ µ ) that refines( f, µ ) by making disks subordinate to T − disjoint. In particular, we may apply the theoryof measurable partitions and conditional measures to ˇ f . However, we cannot compute theentropy of ˇ f using the Rokhlin formula as in [Du4], because constructing invariant partitionsrequires invertible maps. For entropy we consider finally the natural extension of ˇ f . Happily,the natural extension of ˇ f turns out to be isomorphic to that of f ( §
7; see also the figure onp.27 for a synthetic picture of the relationship between these spaces). Hence h ( f, µ ) = h ( ˇ f , ˇ µ ).Once we know that h ( f, µ ) = log λ , the fact that µ is a measure of maximal entropy followsfrom Gromov’s inequality [Gr].The product structure of the natural extension of µ follows from the above analysis andthe analytic continuation property of the disks subordinate to strongly approximable wo-ven/laminar currents ( § Example . Let f : C → C be a monomial map with small topological degree, that is of theform ( z, w ) ( z a w b , z c w d ) where the matrix A = (cid:0) a bc d (cid:1) ∈ GL ( Z ) satisfies | det( A ) | < ρ ( A ),e.g. A = ( n
11 1 ) with n ≥
3. In this case the dynamics of f is well known and the uniquemeasure of maximal entropy is the Lebesgue measure on the unit torus {| z | = | w | = 1 } , whichis a totally invariant subset. Notice also that for this mapping, unstable manifolds do notdepend on histories. Example . Let f be a complex H´enon map of degree d ≥
3, of the form f ( z, w ) =( aw + p ( z ) , az ). Consider now an integer 1 < e < d and for b ≪ a consider the map f b ( z, w ) = ( aw + p ( z ) , az + bz e ). It is easy to prove that f b is algebraically stable in C , with λ = d and λ = e . Now there exists a large bidisk B where f b is an H´enon-like map in thesense of [Du1], in particular injective. So it has a unique measure of maximal entropy log d in B , and it is easy to prove that this measure is actually µ = T + ( f b ) ∧ T − ( f b ), where T ± ( f b )are the global currents constructed above.So in this case most preimages of points in Supp( µ ) actually escape Supp( µ ) and thereforedo not contribute to the dynamics in the natural extension. We see with this example thatit is insufficient to prove a statement like: “for a disk ∆ ⊂ f n ( L ), we have at least (1 − ε ) λ n contracting inverse branches”, since the remaining histories could have full measure in thenatural extension. Our process of selecting distinguished inverse branches in Proposition 5.1will give a way of identifying the “good” preimages.In view of these examples, one may wonder whether there exists a rational map with smalltopological degree such that the number of preimages of points on the support of µ is aconstant strictly between 1 and λ ( f )?Likewise, is there an example where the number of preimages of points on the support of µ is essentially non-constant? By “essentially” we mean that the degree does not only vary on aset of zero measure. Notice that this is not incompatible with ergodicity or mixing; indeed itis easy to construct unilateral subshifts of finite type with this property: consider for instancethe subshift on two symbols 0
1, 1 The negative exponent
In this section we study the contraction properties along T − to estimate the negativeexponent. The main lemma is the following. Recall that T + ∧ τ denotes the transversemeasure induced by T + on τ . Lemma 4.1 (`a la Lyubich) . Let L = { D t , t ∈ τ } be a flow box subordinate to T + . Thenfor every ε > , there exists a positive constant C ( ε ) and a transversal τ ( ε ) ⊂ τ such that M ( T + ∧ τ ( ε )) ≥ (1 − ε ) M ( T + ∧ τ ) and ∀ n ≥ , ∀ t ∈ τ ( ε ) , Area( f n ( D t )) ≤ C ( ε ) n λ n λ n Proof.
The method is similar to [Du4], though it requires substantial adaptation. We freelyuse the structure properties of strongly approximable laminar currents; the reader is referredto [Du4, Du3] for more details.It is enough to prove that for every α and every fixed n , the transverse measure of the setof disks D t such that Area( f n ( D t )) > α is smaller than λ n αλ n . Indeed if this is the case, thenfor every n ≥ c > T + ∧ τ ) (cid:18)(cid:26) t, Area( f n ( D t )) > cn λ n λ n (cid:27)(cid:19) < cn , Lyapunov exponents are well-defined only when (H3) is satisfied: of course here we mean item i. ofTheorem B. EOMETRIC CURRENTS AND ERGODIC THEORY 15 and it will suffice to sum over all integers n and adjust the constant c to get the conclusionof the lemma.We first need to analyze the action of f on the transverse measure. This is a local problem.Recall that T + gives no mass to the critical set [DDG1], and consider an open set U suchthat f : U → f ( U ) is a biholomorphism. If τ is a transversal to some flow box L containedin U , then f ( τ ) is a transversal to f ( L ) and from the invariance relation f ∗ T + = λ T + weinfer that M ( T + ∧ f ( τ )) = λ M ( T + ∧ τ ).Consider our original flow box L and fix n and α . Let τ α ⊂ τ be the set of disks such thatArea f n ( D t ) > α , and L α be the corresponding flow box. Sliding τ along the lamination anddiscarding a set of transverse measure zero if necessary, we can arrange that- τ is contained in a holomorphic disk transverse to L ;- τ ∩ I ( f n ) = ∅ and τ ∩ C ( f n ) is a finite set;- for every p ∈ τ α , p is the unique preimage of f n ( p ) in τ α .We want to estimate the transverse measure of τ α . We exhaust τ α by compact subsets τ ′ α ,such that τ ′ α ∩ C ( f n ) = ∅ , and for simplicity rename τ ′ α into τ α . Observe that f n ( L α ) iscontained in T + so its total mass relative to T + , M ( T + ∧ f n ( L α )) is bounded by 1. On theother hand we will prove that M ( T + ∧ f n ( L α )) ≥ αλ n λ n M ( T + ∧ τ α ) , hence giving the desired result. In constrast to the birational case ( λ = 1), we have to takeinto account the fact that f n ( L α ) will overlap itself, in a way controlled by the topologicaldegree λ n .To give the idea for the rest of the proof, we first consider a model situation: imagine asingle disk D which admits a partition into λ n pieces D i such that f n ( D i ) has area greaterthan α . Then f n ( D ) = S f n ( D i ) has area greater than αλ n /λ n because ∀ p ∈ f n ( D ) , { i, p ∈ f n ( D i ) } ≤ λ n . The following computation is a “foliated” version of this counting argument.If p ∈ L \ C ( f n ), f n is a biholomorphism in some neighborhood N of p . If D p denotesthe disk of L through p , which is subordinate to T + , then f n ( D p ∩ N ) is a disk subordinateto T + . Recall from [Du3] that disks subordinate to T + do not intersect (i.e. they overlapwhen they intersect), so there is an unambiguous notion of leaf subordinate to T + (union ofoverlapping subordinate disks), and a disk subordinate to T + is contained in a unique leaf.Notice that for every t ∈ τ α , D t ∩ C ( f n ) is an at most countable number of points. Since f n ( D t ) has area greater than α , we can remove a small neighborhood N of C ( f n ) such thatfor every t , Area( f n ( D t \ N )) > α . We further assume that N ∩ τ α = ∅ . Now if p ∈ f n ( L \ N ),locally there is a unique disk through p , subordinate to T + (namely f n ( D q ∩ N ( q )) where q ∈ L \ N , is any preimage of p and N ( q ) is a small neighborhood of q ), and we conclude that f n ( L \ N ) is a piece of lamination subordinate to T + . If p ∈ f n ( L \ N ), we denote by L p theleaf subordinate to T + through p .Consider the closed set f n ( τ α ). This is locally a transversal to the lamination f n ( L \ N ),but globally it can intersect a leaf many times. Its total transverse mass is λ n M ( T − ∧ τ α ).Recall that every p ∈ f n ( τ α ) admits a unique preimage q ∈ τ α , and f n ( D q \ N ) is a pieceof a holomorphic curve through p of area greater than α , that we will denote by ∆ p . Byconstruction, a point in f n ( L \ N ) belongs to at most λ n such ∆’s. Let dA L ( x ) denote area measure along the leaf L . We have αλ n M ( T + ∧ τ α ) = α M ( T + ∧ f n ( τ α ))(4) ≤ Z f n ( τ α ) Area(∆ p ) d ( T + ∧ f n ( τ α ))( p )= Z f n ( τ α ) Z L p ∆ p ( x ) dA L p ( x ) ! d ( T + ∧ f n ( τ α ))( p ) . Now, take a partition of f n ( L \ N ) into finitely many flow boxes, and in each flow box,project f n ( τ α ) on a reference transversal τ ref . For simplicity, we will assume that there is onlyone such flow box. The general case follows easily. We can decompose f n ( τ α ), up to a set ofzero transverse measure, into at most countably many subsets τ i intersecting each leaf in asingle point in the flow box, so that we get an injective map h i : τ i → τ ref . Since the transversemeasure is by definition invariant under holonomy, ( h i ) ∗ ( T + ∧ τ i ) = ( T + ∧ τ ref ) | h i ( τ i ) . We canthus resume computation (4) as follows: Z f n ( τ α ) Z L p ∆ p ( x ) dA L p ( x ) d ( T + ∧ f n ( τ α ))( p ) = X i Z τ i Z L p ∆ p ( x ) dA L p ( x ) d ( T + ∧ τ i )( p )= Z τ ref Z L q X i ∆ h − i ( q ) ( x ) dA L q ( x ) d ( T + ∧ τ ref )( q ) ≤ Z τ ref Z L q λ n S i ∆ h − i ( q ) ( x ) dA L q ( x ) d ( T + ∧ τ ref )( q )= λ n Z τ ref Area [ i ∆ h − i ( q ) ! d ( T + ∧ τ ref )( q )= λ n M T + ( f n ( L \ N )) ≤ λ n , where the inequality on the third line comes from the fact that a given point belongs to at most λ n disks ∆. We conclude that M ( T + ∧ τ α ) ≤ λ n /αλ n which was the desired estimate. (cid:3) Proof of item i. in Theorem B.
The conclusion now follows directly from [Du4, p.236]. Herewe give a simpler argument that avoids using the Birkhoff Ergodic Theorem. This shows thattaking sets of density 1 in [Du4] was superflous.Since µ is the geometric intersection of T + and T − , we can fix a finite disjoint union A = L ∪ · · · ∪ L N of flow boxes L i for T + such that µ ( L ∪ · · · ∪ L N ) ≥ − ε . For each p ∈ L i let D p be the disk of L i through p , and let e s ( p ) be the unit tangent vector to D p at p .By the previous lemma, we may discard from A a set of plaques of small transverse measure(hence of small µ measure) to arrange that Area( f n ( D p )) ≤ c ( ε ) n λ n λ n for every p ∈ A and every n ≥
1. By slightly reducing the disks (hence losing one more ε of mass) and applying theBriend-Duval area-diameter estimate [BrDu], we further arrange that the diameter of f n ( D p )is controlled by C ( ε ) nλ n/ /λ n/ . Here, the constant depends on the geometry of the disks D p , hence ultimately on ε since the disks in L ∪ · · · ∪ L N have bounded geometry. So for n large enough, f n ( D p ) is contained in a single coordinate chart of X , and we infer that thederivative df n ( e s ( p )) has norm controlled by C ( ε ) nλ n/ /λ n/ . Thus(5) lim sup n →∞ n log | df n ( e s ( p )) | ≤ − log( λ /λ )2 EOMETRIC CURRENTS AND ERGODIC THEORY 17 for all p ∈ A . Letting ε →
0, we obtain the same inequality holds for µ -almost every p . (cid:3) Distinguished inverse branches
In this section we study the dynamics along the current T − . This will lead to the estimateon the positive exponent and lay the groundwork for computing entropy and proving localproduct structure. Our goal is to prove the following assertion about backward iteration onwhich nearly all subsequent results depend. Proposition 5.1.
For any ε > , there exists a family Q of disjoint cubes, a current T −Q ≤ T − (resp. T + Q ≤ T + ) marked and uniformly woven (resp. uniformly laminar) in each Q ∈ Q ,and a set of distinguished histories such thati. M ( T + Q ˙ ∧ T −Q ) ≥ − ε ;ii. the set of distinguished histories ( x j ) with x ∈ Supp T + Q ˙ ∧ T −Q has measure ≥ − ε in thenatural extension ˆ µ of µ ;iii. if f − n is a distinguished inverse branch of f n along a disk D in the web supporting T −Q ,then the derivative of f − n is controlled by C ( ε ) n/λ n/ ;iv. for every disk D in the web supporting T −Q , there exists at least one compatible sequenceof distinguished inverse branches along D . The proof of the bound in iii. originates in [Du4] and [dT1], but to allow for furtherapplications, we undertake a more elaborate analysis of µ . The exact meaning of the word“distinguished” appearing throughout the statement will be made clear during the proof. Proof of Proposition 5.1.
For a generic hyperplane section L , S k := λ − k [ f k ( L )] convergesto T − . As described earlier, we choose a generic subdivision e Q by cubes and extract from S − k := λ − k [ f k ( L )] a (uniquely marked) uniformly woven current S − k, e Q which is the restrictionof S k to disks (graphs over one projection in Q ) of area not greater than 1 / ε be a small positive number. Fix the subdivision e Q as above, together with a corre-sponding family Q of slightly smaller concentric cubes, homothetic to those of e Q by a factor(1 − δ ). Then for an appropriate choice of e Q ,(6) µ ( e Q \ Q ) ≤ − (1 − δ ) ) . See [Du2, Lemma 4.5] for a proof.Now remove from S − k, e Q all components not intersecting Q , and denote by S − k, Q the remainingcurrent. Observe that S − k, e Q | Q = S − k, Q | Q . As expained in § S − k, Q are markedby measures on Z ( e Q , /
2) whose masses are bounded uniformly in k .For reasons that will become clear in Theorem 6.1 below (roughly speaking, to get someinvariance for the web supporting T − ), we do some further averaging, by setting T − k = k P ki = ⌊ k/ ⌋ S − i , where ⌊·⌋ denotes the integer part function. Observe that T − k → T − as k → ∞ , and that restricting T − k to the good components (as defined in § T − k , we getthe uniformly woven current T − k, Q = k P ki = ⌊ k/ ⌋ S − i, Q .Since the transverse measures of T − k, Q have uniformly bounded mass, we may extract astrongly converging subsequence of T − k, Q . The limiting current T −Q is automatically uniformlywoven (in each cube). We similarly construct currents T + Q associated to backward images ofa generic line, though in this case, the extra averaging step, while harmless, is not necessary. Recall that the T ±Q are actually defined in the slightly bigger subdivision e Q . If the size ofthe cubes is small enough, then(7) M (cid:0) T + ∧ T − − ( T + Q ˙ ∧ T −Q ) (cid:1) will be small. So in the following we fix e Q ⊃ Q so that the sum of the error terms in (6) and(7), together with the loss of mass coming from a neighborhood of the set of points wherethe two projections are not transverse fibrations is less than ε/
10. For technical reasons, wefurther require that µ ( ∂ Q ) = 0.By Lemma 1.5, we can define the measures µ k = T + ∧ T − k and T + ∧ T − k, Q for every k ≥ µ k tends to 1 for cohomological reasons. We claim that in fact µ k → µ as k → ∞ .Indeed, the wedge product of uniformly woven currents is geometric, so from Lemma 1.4 wehave for large k (depending only on Q , hence on ε ) that(8) M (cid:16) ( T + ∧ T − k, Q ) | Q (cid:17) ≥ M (cid:16) ( T + Q ∧ T − k, Q ) | Q (cid:17) = M (cid:16) ( T + Q ˙ ∧ T − k, Q ) | Q (cid:17) ≥ − ε , More generally, if k j is any sequence such that T − k j , Q strongly converges to some T Q , thenlim T + Q ˙ ∧ T − k j , Q ≥ T + Q ˙ ∧ T −Q . So any cluster value of µ k is larger than T + Q ˙ ∧ T −Q , whence µ k → µ .Denote by µ k, Q the measure T + Q ∧ T − k, Q = T + Q ˙ ∧ T − k, Q . Backward contraction for µ k, Q . Fix an integer n ≥
1. Then for generic L and any k ≥ n , we have that f n : f k − n ( L ) → f k ( L ) is 1-1 outside some finite set. So for every disk D ֒ → f k ( L ), f n admits a unique (“distinguished”) inverse branch f − n : D → f k − n ( L ). Sincethe area of f k ( L ) is no greater than Cλ k for some C >
0, we have (cid:26) plaques D of S − k, Q , s.t. Area( f − n ( D )) ≥ An λ n (cid:27) ≤ Cλ k An , and thus (cid:26) plaques D of S − k, Q , s.t. ∃ n ≤ k, Area( f − n ( D )) ≥ An λ n (cid:27) ≤ k X n =1 Cλ k An ≤ C ′ λ k A Discarding the plaques in the latter set from S − k, Q , we get a new uniformly woven current,that we denote by S − k, Q ( A ). In terms of the transverse measure, we have removed at most C ′ λ k /A Dirac masses . Hence(9) M (cid:16) m ( S − k, Q ) − m ( S − k, Q ( A )) (cid:17) ≤ CA .
We then form the current T − k, Q ( A ) = k P ki = ⌊ k/ ⌋ S − i, Q ( A ) (which also satisfies (9)). Ex-tracting a strongly convergenct subsequence, we get a current T −Q ( A ) ≤ T −Q . This family ofcurrents increases to T −Q as A increases to infinity (technically we need to take a sequenceof A ’s and a diagonal extraction), hence by geometric intersection T + Q ∧ T −Q ( A ) increases to T + Q ∧ T −Q . We fix A so that M ( T + Q ∧ T −Q ( A )) ≥ − ε/
2. Relabeling, we now use T −Q to denote T −Q ( A ) and T − k, Q to denote T − k, Q ( A ). We will show that the conclusions of Proposition 5.1 holdfor this current T −Q and the current T + Q constructed above. It is important here that the transverse measure is a sum of Dirac masses on disks, not on arbitrary chains.
EOMETRIC CURRENTS AND ERGODIC THEORY 19 If n is fixed, then as soon as k/ ≥ n , the “distinguished” preimage of each plaque of T − k, Q under f n has small area by construction. By lower semicontinuity (Lemma 1.4), we get thatfor large k , M ( µ k, Q ) ≥ − ε/ f − n along the plaques of T − k, Q is small in Q . To be more specific,if Q ∈ Q and D is a plaque of T − k, Q in e Q , the modulus of the derivative of f − n along D ∩ Q will be controlled by C ( ε ) n/λ n/ . Indeed, recall that D has area ≤ / D ∩ Q in D are bounded from below by a constant depending only on the geometry of Q and e Q ,hence ultimately on ε . The estimate on the derivative then follows from the original estimateof Briend and Duval. Distinguished inverse branches.
For each Q ∈ Q and each plaque D of T − Q in Q , D is theHausdorff limit of a sequence of disks D k with the property that for every 1 ≤ n ≤ k/ f n admits a natural inverse branch f − n over D k , with a uniform control on the derivative along D k , of the form(10) | d ( f − n | D k ) | ≤ C ( ε ) n/λ n/ . As already explained (see the examples in §
3) it will be important to isolate a set of“meaningful” histories in the natural extension. Here is the crucial definition.
Definition 5.2.
Let D be a disk subordinate to T −Q . We call an inverse branch f − n of f n along D distinguished if there exists a sequence of disks D k subordinate to T − k, Q converging to D such that the natural branches f − n | D k converge normally to f − n | D .We say that a history ( x − j ) ≤ j ≤ n of length n is Q -distinguished if x ∈ Supp( T + Q ˙ ∧ T −Q ) and x − n = f − n ( x ) for some distinguished inverse branch f − n . An infinite history ( x − i ) i ≥ is Q -distinguished if all its subhistories of length n are Q -distinguished. When there is no danger of confusion we will omit the ‘ Q ’ in Q -distinguished; more gen-erally, “distinguished” will stand for “ Q -distinguished for some Q ”. Taking normal limits ofthe f − n | D k , we see that f n admits distinguished inverse branches on every disk suboordinateto T −Q . By diagonal extraction, we further find that every x ∈ Supp( T + Q ˙ ∧ T −Q ) admits adistinguished (full) history. Moreover, attached to every Q -distinguished history of x , thereis a disk D ∋ x subordinate to T −Q and a sequence of distinguished inverse branches f − n | D ,with f − n ( x ) = x − n , compatible in the sense that f ◦ f − n − = f − n . Since the estimate in (10)is uniform in k we have the estimate | d ( f − n | D ) | ≤ C ( ε ) n/λ n/ . This proves items iii. and iv. of Proposition 5.1.It remains to show that distinguished histories are overwhelming in the natural extension of µ . Let ˆ X dist Q ⊂ ˆ X be the set of distinguished histories ( x j ) of points x ∈ Supp( µ Q ); likewise,let ˆ X dist be the increasing union of ˆ X dist Q as the diameter of the cubes in Q goes to zero. Weprove that ˆ µ ( ˆ X dist ) = 1 by proving that ˆ µ ( ˆ X dist Q ) ≥ − ε .Let ˆ X dist − N, Q ⊂ ˆ X consist of histories of points x ∈ Supp µ that are distinguished up to time − N . This is a decreasing sequence of subsets of ˆ X , and T N ≥ ˆ X dist − N, Q = ˆ X dist Q . It is enoughto prove that ˆ µ ( ˆ X dist − N, Q ) ≥ − ε for all N ≥
1. Now, ˆ µ ( ˆ X dist − N, Q ) = ˆ µ (cid:0) ˆ f N ( ˆ X dist − N, Q ) (cid:1) and by definition ˆ f N ( ˆ X dist − N, Q ) is the set of sequences ( x n ) such that x is a cluster value of a sequence f − N ( x ( k ) N ), with Supp( S − k, Q ) ∋ x ( k ) N → x N .Recall that the measure µ k, Q has mass larger than 1 − ε , and satisfies(11) µ k, Q := T + Q ∧ T − k, Q ≤ T + ∧ T − k, Q = 2 k k X i = ⌊ k/ ⌋ T + ∧ S − i, Q . We come now to the crucial point. Since there is a natural f − N on f k ( L ) for k > N , we mayconsider the measure ( f − N ) ∗ µ k, Q , which has mass larger than 1 − ε . From (11) we infer that( f − N ) ∗ µ k, Q ≤ T + ∧ T − k − N + σ k , where σ k is a signed measure of total mass O ( Nk ). Considerany cluster value of this sequence of measures as k → ∞ and denote it by ( f − N ) ∗ µ Q (thisnotation is convenient but somewhat improper). Then ( f − N ) ∗ µ Q ≤ µ because µ k → µ and M (( f − N ) ∗ µ Q ) ≥ − ε . Now if ( x n ) ∈ ˆ X is any sequence such that x ∈ Supp(( f − N ) ∗ µ Q ),then by definition ( x n ) ∈ (cid:0) ˆ f N ( ˆ X dist − N, Q ) (cid:1) (note that the tail ( x j ) j< − N of a history in ˆ X dist − N, Q isarbitrary). Henceˆ µ (cid:16) ˆ f N ( ˆ X dist − N, Q ) (cid:17) ≥ ˆ µ (cid:0) ˆ π − (Supp(( f − N ) ∗ µ Q ) (cid:1) = µ (Supp(( f − N ) ∗ µ Q )) ≥ − ε. This finishes the proof of the proposition. (cid:3)
Remark . i. From the first part of the proof we could directly deduce the existence ofthe positive exponent (expansion in forward time), in the spirit of [dT1]. However, toobtain more complete results, we first construct the tautological extension.ii. The laminarity of T + is only used in the proof to get lower semicontinuity properties ofthe wedge products. So if, for instance, T + has continuous potential, we can drop thelaminarity assumption and get the same conclusion. Notice additionally in this case thatthe wedge product T + ∧ T − is “semi-geometric” in the sense that it is approximated frombelow by T + ∧ T −Q (see the proof of [Du2, Remarque 4.6] or [Du4, Remark 5.3]). This isthe setting of [dT1], and might be useful for further applications.6. The tautological extension
So far we have constructed a family of marked uniformly woven currents T −Q = P T − Q ,increasing to T − , with the property that for any disk appearing in the markings, f n admitsexponentially contracting inverse branches. We say that such disks are subordinate to T − .If D is a disk subordinate to T − , we define the measure T + ˙ ∧ D as the increasing limit of themeasures T + Q i ˙ ∧ D for our choice of increasing subdivisions Q i . Since T + ∧ T − is a geometricintersection, if D is a generic (relative to the marking) disk subordinate to T − , then T − ∧ [ D ]is well defined and T + ∧ [ D ] = T + ˙ ∧ D .We now construct the tautological extension ( ˇ X, ˇ µ, ˇ f ) of ( X, µ, f ). This is roughly speakingthe “smallest” space where the unstable leaves become separated. It will be realized as adisjoint union of flow boxes, foliated by the lifts of the disks subordinate to T − (whichplay the role of unstable manifolds). The marking data give us a lift of T − to a “laminar This space does not depend canonically on (
X, µ, f ). On the other hand it depends canonically on µ viewedas a geometric intersection of marked woven currents. We make no attempt to connect the flow boxes, as this would certainly lead to unwelcome topologicalcomplications.
EOMETRIC CURRENTS AND ERGODIC THEORY 21 current” on ˇ X . Hence we get a lift ˇ µ of µ , whose conditionals on unstable manifolds are wellunderstood. This will be our main technical step towards the understanding the conditionalsof ˆ µ on unstable manifolds in the natural extension. As suggested by the referee, it might bepossible to analyze the conditionals of ˆ µ directly from µ but we don’t know how to do it. Theorem 6.1.
There exists a locally precompact and separable space ˇ X , which is a countableunion of compatible flow boxes, together with a Borel probability measure ˇ µ and a measurepreserving map ˇ f , with the following properties.i. There exists a projection ˇ π : ( ˇ X, ˇ µ ) → ( X, µ ) semiconjugating f and ˇ f , i.e. ˇ π ◦ ˇ f = f ◦ ˇ π .ii. ˇ X admits a measurable partition ˇ D whose atoms ˇ D (ˇ x ) , ˇ x ∈ ˇ X , are mapped homeomor-phically by ˇ π to onto disks subordinate to T − .iii. The conditional measure of ˇ µ on almost any atom ˇ D (ˇ x ) is induced by the current T + asfollows: it is equal up to normalization to (cid:0) (ˇ π | D (ˇ x ) ) − (cid:1) ∗ ( T + ˙ ∧ D ) where D = ˇ π ( ˇ D (ˇ x )) .Proof. As before, let Q be one among a fixed increasing sequence ( Q i ) of subdivisions with µ ( ∂ Q i ) = 0. The current T Q is marked by a measure on the disjoint union ` ˜ Q ∈ ˜ Q Z ( ˜ Q, / / § Z ( Q ) of Z ( Q ) are pairs ( x, D ) with x ∈ D ∈ Z ( Q ), and that theprojection ˇ π : ˇ Z ( Q ) → Z ( Q ) is given by ˇ π ( x, D ) = x .The principle of the proof is quite simple. Each D ∈ Supp m ( T − Q ) admits a natural lift ˇ D to ˇ Z ( Q ), and T + induces a measure on ˇ D according to the formula T + ˙ ∧ ˇ D := (cid:0) (ˇ π | ˇ D ) − (cid:1) ∗ ( T + ˙ ∧ [ D ]) , Averaging with respect to the markings then gives a measureˇ µ Q := Z ( T + ˙ ∧ ˇ D ) d ( m ( T − Q ))( D )that projects on µ Q . We call ˇ µ Q the tautological extension of T + ˙ ∧ T − Q . What remains is tomake sense of the “increasing limit” ˇ µ of P Q ∈Q i µ Q as i → ∞ . In particular, we need toconstruct the space ˇ X that carries ˇ µ , and then show that ˇ µ is invariant under some naturallyassociated map ˇ f .To construct ˇ X we recall from Lemma 1.2 that for generic subdivisions, almost all chainsappearing in the markings m ( T −Q ) are disks transverse to the boundary (this is an opensubset of chains). Let O ⊂ Supp( m ( T −Q )) be the relatively open, full measure subset ofboundary-transverse disks. Set E = O and let ˇ E be the tautological bundle over E .Now assume that for 1 ≤ j ≤ i − E j and ˇ E j have been constructed. Considerthe marked current T −Q i , and as before denote by O i the open subset of Supp( m ( T −Q i )) madeof boundary-transverse disks. We add to S j ≤ i − E j the smaller set E i ⊂ O i consisting onlyof those disks which have not previously appeared: i.e. E i = O i \ n D : D ⊂ Z ∈ Supp( m ( T −Q j )) o . As is easily verified, E i is open in O i and therefore also in the locally compact metric space ` Q ∈Q i Z ( Q ).Now take ˇ X to be the disjoint union ˇ X = ` i ˇ E i . By Lemma 1.1, there is no folding in m ( T −Q j ) so each ˇ E i is laminated by the disks of E i . We can endow ˇ X with a natural topology by putting the natural topology on each ˇ E j , and declaring that each ˇ E j is open and closed.This topology is even induced by a metric where each of the ˇ E i is bounded, and at definite(positive) distance from the others. This makes ˇ X a locally precompact and separable space.We could also take its completion to get a locally compact and separable space but we willnot need it. Each E i is naturally partitioned by the disks of E i , giving rise to the measurablepartition ˇ D of ii. , and can be covered by a countable family of compatible flow boxes.To define the measure ˇ µ , we note that if D ⊂ D ′ with D ∈ Z ( Q ) (resp D ′ ∈ Z ( Q ′ )), thenthere is a natural inclusion ˇ D ֒ → ˇ D ′ . Hence we can view the tautological extension of T + ˙ ∧ T −Q i as a measure ˇ µ Q i supported on E ∪ · · · ∪ E i (rather than Z ( Q i )). This defines an increasingsequence of measures in ˇ X . The limit ˇ µ is a Borel probability measure. The conditionalof the measure ˇ µ Q i on a disk D of ˇ E j , is by definition induced by T + (and independent of i ≥ j ), so statement iii. follows.Finally, we seek to construct the measurable map ˇ f projecting onto f and leaving ˇ µ in-variant. We want to define ˇ f as follows: ˇ f ( x, D ) = ( x ′ , D ′ ) if f ( x ) = x ′ and germ x ′ ( f ( D )) =germ x ′ ( D ′ ). The details of this are a little involved, however.To begin with, let us recall some notation from Proposition 5.1. In the course of provingthis result, we introduced currents T − k = k P kj = k/ λ j S j together with the restriction T − k, Q of T − k to disks which were ‘good’ relative to Q . We also had measures µ k = T + ∧ T − k and µ k, Q = T + ∧ T − k, Q (note the slight change in the definition of the latter).It is immediate from the definitions that the difference σ k := f ∗ µ k − µ k has mass no greaterthan 4 /k . Another useful observation is that, for purposes of comparing the various measureswe have defined, we can be a little flexible concerning their domains. Since µ k gives no mass topoints (Lemma 1.5) and µ k, Q is concentrated on countably many disks of Z ( Q ) with discreteintersections, we can lift µ k, Q canonically to a measure ˇ µ k, Q on ˇ Z ( Q ). So we may regard µ k, Q as a measure on the cubes of Q or alternatively as a measure on the the tautological bundleˇ Z ( Q ). In a similar vein, we may regard ˇ µ | ˇ E ∪···∪ ˇ E i as a measure on ˇ Z ( Q i ) rather than ˇ X . Lemma 6.2.
Let ˇ ν Q i be any cluster value of the sequence of measures ˇ µ k, Q i . Then ˇ ν Q i − ˇ µ Q i is a signed measure with M (ˇ ν Q i − ˇ µ Q i ) = ε ( Q i ) where ε ( Q i ) depends only on i and tends to zero as i → ∞ .Proof of Lemma 6.2. Since T + ≥ T + Q i , we haveˇ µ k, Q i ≥ Z ( T + Q i ˙ ∧ ˇ D ) d ( m ( T − k, Q i ))( D ) . Taking cluster values on both sides, and using lower semicontinuity (Lemma 1.4) we inferthat ˇ ν Q i ≥ Z ( T + Q i ˙ ∧ ˇ D ) d ( m ( T −Q i ))( D ) . On the other hand,ˇ µ Q i = Z ( T + ˙ ∧ ˇ D ) d ( m ( T −Q i ))( D ) ≥ Z ( T + Q i ˙ ∧ ˇ D ) d ( m ( T −Q i ))( D ) . Note that ˇ µ Q i might exceed ˇ µ Q i − i not only on E i but also on those earlier sets E j that contain disks in O i . EOMETRIC CURRENTS AND ERGODIC THEORY 23
Hence ˇ µ Q i , ˇ ν Q i are both measures with at most unit mass, and both are bounded below by ameasure which, by geometric intersection, has mass at least 1 − ε ( Q i ). (cid:3) Continuing with the proof of the theorem, we let A k, Q i = Supp T − k, Q i ∩ f − (Supp T − k, Q i ).That is, A k, Q i consists of points that ‘go from large disks to large disks.’ Since f ∗ µ k = µ k + σ k and f is essentially 1-1 on Supp T − k , we have that A k, Q i has almost full mass: µ k ( A k, Q i ) ≥ µ k (Supp T − k, Q i ) − µ k (Supp T − k − Supp T − k, Q i ) − k . By geometric intersection, the second term on the right hand side is of the form ε ( k, Q i ) withlim k →∞ ε ( k, Q i ) = ε ( Q i ) and ε ( Q i ) → i → ∞ . So since µ k, Q i = µ k | Supp T k, Q i , we inferthat(12) M (cid:16) µ k, Q i − µ k, Q i | A k, Q i (cid:17) ≤ M (cid:16) µ k − µ k, Q i | A k, Q i (cid:17) ≤ ε ( k, Q i ) + 4 k . In the same way we obtain that(13) M (cid:16) µ k, Q i − f ∗ µ k, Q i | A k, Q i (cid:17) ≤ ε ( k, Q i ) + O ( 1 k ) . Lifting to the tautological bundle, we get a set ˇ A k, Q i . Consider a cluster value ν A of thesequence of measures ˇ µ k, Q i | ˇ A k, Q i . From Lemma 6.2 and (12), and since there is no loss ofmass in the boundary, we get that M ( ν A − ˇ µ Q i ) ≤ M ( ν A − ˇ ν Q i ) + M (ˇ ν Q i − ˇ µ Q i ) = ε ( Q i ) , where ˇ ν Q i is a cluster value of the sequence of measures ˇ µ k, Q i .Now set ˇ A Q i := lim sup k →∞ A k, Q i . Since this contains Supp( ν A ); we infer that ˇ µ Q i ( ˇ A Q i ) ≥ − ε ( Q i ). We claim moreover that there is a well-defined map ˇ f i : ˇ A Q i → Z ( Q i ) givenby f i ( x, D ) = ( f ( x ) , D ′ ) where D ′ ∈ Z ( Q i ) coincides with f ( D ) near f ( x ). To see that thisworks, observe that by definition of ˇ A Q i , there is a sequence of pointed disks ( x k , D k ) → ( x, D )such that D k is subordinate to T + k, Q i and f ( D k ) coincides near f ( x k ) with some other disk D ′ k subordinate to T + k, Q i . Taking D ′ to be a cluster value of the ( D ′ k ) k ∈ N , we see that ˇ f i isindeed well-defined. Lemma 6.2 and (13) tell us additionally that(14) M (cid:16) ( ˇ f i ) ∗ (ˇ µ Q i | ˇ A Q i ) − ˇ µ Q i (cid:17) = ε ( Q i ) . Rephrasing the preceding construction in terms of ˇ X , we have constructed a set ˇ A Q i ⊂ ˇ E ∪ · · · ∪ ˇ E i , with ˇ µ -mass ≥ − ε ( Q i ), together with a map ˇ f i : A Q i → ˇ E ∪ · · · ∪ ˇ E i , thatcoincides with the action of f on the space of germs. If, when refining the subdivision, weare careful to extract our subsequences from those chosen for earlier subdivisons, then wewill obtain an increasing sequence of subsets ( ˇ A Q i ), with the compatibility (say Q j is finerthan Q i ) ˇ f j | ˇ A Q i = ˇ f i . So the maps ˇ f i piece together to form a single map ˇ f defined on a fullmeasure subset of ˇ X . Furthermore, since M (cid:16) ( ˇ f i ) ∗ ˇ µ | ˇ A Q i − ˇ µ | ˇ E ∪···∪ ˇ E i (cid:17) = ε ( Q i )we infer that ˇ µ is ˇ f -invariant. (cid:3) The following proposition clarifies the relationship between iteration on ˇ X and the con-struction of the previous section. We say that a disk has size ≥ r if it belongs to Z ( Q ) forsome cube Q of size ≥ r . Proposition 6.3.
For every fixed positive integer ℓ , there exists a set ˇ A ε ( ℓ ) ⊂ ˇ X of ˇ µ -measure ≥ − ε such that if ˇ x = ( x, D ) ∈ ˇ A ε ( ℓ ) then ˇ f ℓ (ˇ x ) has the following properties:i. f ℓ ( D ) coincides near f ℓ ( x ) with a disk of size ≥ r ( ε ) > ;ii. f ℓ : D → f ℓ ( D ) is univalent, with derivative larger than C ( ε ) λ ℓ/ ℓ − ;iii. the orbit segment x, f ( x ) , . . . , f ℓ ( x ) is distinguished.Proof. Replace the set A k, Q i used in the previous proof with the analogous set A k, Q i ( ℓ ) ofpoints x ∈ Supp T − k, Q i such that f ℓ ( x ) ∈ Supp T − k, Q i . Define A Q i ( ℓ ) = lim sup( A k, Q i ( ℓ )) andconsider as before the tautological bundles over these sets. As we now explain, it suffices totake ˇ A ε ( ℓ ) = ˇ A Q i ( ℓ ) for large enough i .We first need to check that ˇ f ℓ (ˇ x ) is well defined for almost every point in ˇ A Q i ( ℓ ). Thepoint is that there is a piece of disk in Z ( Q i ) sent by f ℓ to a piece of disk of Z ( Q i ), howeveralong the branch x, f ( x ) , . . . , f ℓ ( x ) the disk can become small. Nevertheless if x / ∈ Crit( f ℓ ), f q is locally invertible at x for 1 ≤ q ≤ ℓ , so all the germs f q ( D ) are traced on disks of some , possibly much smaller, size Q i ′ depending on ℓ . The same holds for disks subordinateto T − k, Q i that approximate D . We conclude that ℓ successive iterates of ˇ f are defined at ˇ x .Now, the conclusions of the lemma follow easily from the analysis of distinguished inversebranches in the proof of Proposition 5.1. The only point that needs explanation is ii. If( x, D ) ∈ ˇ A Q i ( ℓ ), then by construction, ( x, D ) is the limit of a sequence of pointed disks( x k , D k ) with (cid:13)(cid:13) df ℓx k ( e (ˇ x k )) (cid:13)(cid:13) ≥ C ( Q i ) λ ℓ/ ℓ − , where e (ˇ x k ) is the unit tangent vector to D k at x k . Recall from Lemma 1.1 that since the current T − is strongly approximable, there isno multiplicity in the convergence of the disks subordinate to T −Q ,k . So if ˇ x k → ˇ x in thisconstruction, then e (ˇ x k ) → e (ˇ x ). It follows for x / ∈ I ( f ℓ ) that df ℓx k ( e (ˇ x k )) → df ℓx ( e (ˇ x )), givingthe desired estimate. We note that since the estimate extends across finite sets, it holds evenat points in I ( f ℓ ). (cid:3) Now we can estimate the positive exponent. For convenience here we take for granted thatˇ µ is ergodic, a fact we will prove in Corollary 7.4 below. Corollary 6.4.
For µ -a.e. x there exist a tangent vector e u at x and a set of integers N ′ ⊂ N of density 1 such that (15) lim inf N ′ ∋ n →∞ n log | df n ( e u ( x )) | ≥ log λ . Given ˇ x = ( x, D ) ∈ ˇ X , we let e (ˇ x ) denote the tangent vector to D at x . The proof makesevident that one can take e u = e (ˇ x ) for ˇ µ a.e. ˇ x . Proof.
We have the following lemma from elementary measure theory (see below for theproof).
Lemma 6.5.
Let ( Y, m ) be a probability space, and ( A n ) n ≥ a collection of sets of measure ≥ − ε . Then for every δ > ε , m ( { y ∈ Y, y ∈ A n for a set of integers n of density ≥ − δ } ) ≥ − εδ . With notation as in Proposition 6.3, letˇ A ε = (cid:8) ˇ x, ˇ x ∈ ˇ A ε ( ℓ ) for a set of integers ℓ of density ≥ − √ ε (cid:9) . technically, we need to avoid the measure zero set of x sent by f q onto boundaries of cubes EOMETRIC CURRENTS AND ERGODIC THEORY 25
By the previous lemma, ˇ µ ( ˇ A ε ) ≥ − √ ε . If ˇ x ∈ ˇ A ε , then (15) holds for a set of integers N ε of density ≥ − √ ε . To conclude, observe that ˇ A ε is an invariant set, so that by ergodicityit has full measure. (cid:3) Proof of Lemma 6.5.
Consider the function ϕ N = P Nn =1 A n (with possibly N = ∞ ). Wehave that 0 ≤ ϕ N ≤ N and R ϕ N dm ≥ (1 − ε ) N . We leave the reader prove that for every δ > ε , m ( B N ) ≥ − εδ where B N = { y, ϕ N ( y ) ≥ (1 − δ ) N } . In particular taking δ close to 1, at this point we conclude that for every fixed C and N largeenough, m ( { ϕ N ≥ C } ) ≥ − ε . In particular for every C m ( { ϕ ∞ ≥ C } ) ≥ − ε , so the setof points belonging to infinitely many A n has measure ≥ − ε .Now the ( B N ) themselves form an infinite collection of sets of measure ≥ − ε/δ so applyingthe same reasoning proves that the set of y belonging to infinitely many B N ’s has measure ≥ − ε/δ which is the desired statement. (cid:3) The natural extension and entropy
In this section we analyze the natural extension of (
X, µ, f ). There are two main steps:prove that the natural extension of (
X, µ, f ) is the same as that of ( ˇ X, ˇ µ, ˇ f ), and analyze theconditional measures of ˆ µ relative to the unstable partition in the natural extension.We first show that different disks subordinate to T − correspond to different histories, asone would expect for unstable manifolds. Proposition 7.1.
Let ˆ x ∈ ˆ X be a ˆ µ -generic point. Then there exists a unique disk D subordinate to some T −Q , together with a sequence of inverse branches f ˆ x, − n defined on D ,such that f ˆ x, − n ( x ) = x − n and f ˆ x, − n contracts exponentially on D .Proof. Proposition 5.1 tells us that distinguished histories have full measure in the naturalextension. So we may assume that ˆ x is Q -distinguished for some Q . We therefore have adisk D ∋ x that is subordinate to T −Q and equipped with a compatible sequence ( f − n ) ofinverse branches such that f − n ( x ) = x − n . Proposition 5.1 further guarantees that f − n isexponentially contracting on D . It remains to show D is unique.Consider the measure µ Q = T + Q ˙ ∧ T −Q . By construction, for each x ∈ Supp( µ Q ) there existsa radius r = r ( x ) such that the disks subordinate to T + Q are submanifolds in B ( x, r ) and getcontracted at uniform exponential speed O ( nλ n /λ n ) under forward iteration. Every point inSupp( µ Q ) has such a local “stable manifold”, which is then unique because T + is stronglyapproximable and laminar [Du3, Theorem 1.1]. Let L s ∩ B ( x, r ) be the stable lamination near x ∈ Supp µ Q , that is, the union of stable manifolds of points in Supp( µ Q ) ∩ B ( x, r ). Sincethe image of a disk subordinate to T + under f (resp. under a branch of f − defined in anopen set) is a disk subordinate to T + , the stable lamination is invariant under the dynamics: f ( L s ∩ B ( x, r )) ⊂ L s .Now suppose for some generic history ˆ x , that D , D are two disks through x satisfyingthe conclusions of the proposition. Then for n large enough, we have f − n ( D j ) ⊂ B ( x − n , r / r = r ( x ). And since x ∈ Supp µ Q ⊂ Supp µ , Poincar´e recurrence for ˆ µ gives aninfinite set S ⊂ N of n such that x − n ⊂ B ( x , r / L is any leaf in L s ∩ B ( x , r ) thatmeets both D and D , then for every n ∈ S , f − n ( L ) ∩ B ( x , r ) is contained in a leaf L ′ intersecting the preimages of either disk f − n ( D j ). Contraction of stable leaves then gives.distance( L ∩ D , L ∩ D ) ≤ Cn (cid:18) λ λ (cid:19) n diameter( L ′ ) = O (cid:18) nλ n λ n (cid:19) . As this is true for all n ∈ S , we conclude that L ∩ D = L ∩ D . Similarly, neither D j canbe entirely contained in the stable leaf through x , because pulling back and pushing forwardwould, in the same fashion, show that the diameter of D j vanishes.Now we can conclude. Leaves in L s ∩ B ( x , r ) accumulate on the one through x . Hencethere are infinitely many such leaves intersecting both D and D . Since the intersectionpoints all lie in D ∩ D , we see that x is an accumulation point of D ∩ D . It follows that D = D ; i.e. the disk D in the proposition is unique. (cid:3) Remark . As is clear from the proof, the uniqueness assertion of the proposition holds for agiven distinguished history ˆ x as long as there are infinitely many n for which x − n ∈ Supp( µ Q ),and we only need f ˆ x, − n to be contracting for these n .The proof also makes clear that the disk D is unique regardless of whether it is subordinateto T −Q . This shows that the web supporting T −Q is essentially independent, along histories inˆ X , of the manner in which it was constructed. It also shows that for almost any ˆ x ∈ ˆ X , thedisk D is the only reasonable candidate for a ‘local unstable manifold’ associated to ˆ x . Wewill therefore refer to such disks as local unstable manifolds and to the resulting partitionsof ˆ µ and ˇ µ as the ‘unstable partition’ of each. We should emphasize, however, that we donot know whether D is the full local unstable set of ˆ x —i.e. whether something like the localunstable manifold theorem holds in the present context. Proposition 7.3.
The natural extension ( b ˇ X, b ˇ µ, b ˇ f ) of ( ˇ X, ˇ µ, ˇ f ) is measurably isomorphic tothat of ( X, µ, f ) .Proof. The natural extension of ( ˇ X, ˇ µ, ˇ f ) is the set of sequences (ˇ x n ) = ( x n , D n ), indexedby Z , with ˇ f (ˇ x n ) = ˇ x n +1 . Observe that if x does not belong to S n ≥ f n ( C ( f n )), then forpositive n , there is a unique germ D − n at x − n such that f n ( D − n ) = D . In particular thewhole sequence ( D n ) is determined by D and ( x n ).Furthermore, since ( b ˇ X, b ˇ µ, b ˇ f ) is an invertible dynamical system projecting onto ( X, µ, f ),the universal property of ( ˆ X, ˆ µ, ˆ f ) gives us an intermediate semiconjugacy η : ( b ˇ X, b ˇ µ, b ˇ f ) → ( ˆ X, ˆ µ, ˆ f ). In explicit terms, η ( x n , D n ) = ( x n ).Since the set of distinguished histories ˆ X dist has full measure in ˆ X , we get that for b ˇ µ -a.e.( x n , D n ), ( x n ) ∈ ˆ X dist . By Proposition 7.1 above, associated to the sequence ˆ x = ( x n ) ∈ ˆ X dist ,there is a unique germ of disk D (ˆ x ) through x which is contracted exponentially in thepast along the branch x − n . The proof will be finished if we show that for b ˇ µ -a.e. ( x n , D n ), D = D (ˆ x ). Indeed, since ( D n ) depends only on D , we will have found an inverse for η .Consider the sets ˇ A ε ( ℓ ) as defined in Proposition 6.3, and define b ˇ A ε ( ℓ ) to be the set ofsequences (ˇ x n ) ∈ b ˇ X with ˇ x − ℓ ∈ ˇ A ε ( ℓ ). We have that b ˇ µ ( b ˇ A ε ( ℓ )) ≥ − ε . Hence by lemma 6.5there is a set b ˇ A ε ⊂ b ˇ X of measure ≥ − √ ε of points belonging to b ˇ A ε ( ℓ ) for infinitely many ℓ .By definition, if ( x n , D n ) ∈ b ˇ A ε , then for infinitely many integers ℓ , the (germ of) disk D iscontracted by the branch of f − ℓ sending D to D − ℓ . By Proposition 7.1 and Remark 7.2 weconclude that D = D (ˆ x ). (cid:3) EOMETRIC CURRENTS AND ERGODIC THEORY 27
Corollary 7.4.
The measure ˇ µ is ergodic under ˇ f , and we have equality between entropies h ( f, µ ) = h ( ˇ f , ˇ µ ) .Proof. Since µ is ergodic, so is ˆ µ , and therefore ˇ µ . Also we have that h ( f, µ ) ≤ h ( ˇ f , ˇ µ ) ≤ h ( ˆ f , ˆ µ ) = h ( f, µ ). (cid:3) From now on, depending on the context, we can think of ( ˆ X, ˆ µ, ˆ f ) as the natural extensionof either ( X, µ, f ) or ( ˇ X, ˇ µ, ˇ f ). Figure 1 illustrates the relationship between the natural andtautological extensions. A disk subordinate to T − and its preimages under ˇ π and π has beenunderlined. The notation ˇ L s , ˇ L u is introduced in the proof of Theorem 8.1 The existence ofthe dashed arrow is ensured by the previous proposition, i.e. by the fact that η is a measurableisomorphism. With notation as above, it is defined by ˆ x ( x , D (ˆ x )).PSfrag replacements W u W s T + T − ˇ L u ˇ L s X ˇ X ˆ X π ˇ π Figure 1.
Schematic picture of the tautological and natural extensions ⋄ Aside: unstable manifolds and the natural extension.
As it is well known, definingunstable manifolds for a non invertible system requires working in the natural extension.The unstable set of ˆ x ∈ ˆ X is W u (ˆ x ) = n ˆ y ∈ ˆ X, lim n →∞ dist( ˆ f − n ( x ) , ˆ f − n ( y )) = 0 o . Undersome hyperbolicity assumptions on ˆ x , W u loc (ˆ x ) projects isomorphically onto a submanifoldembedded in a neighborhood of x ∈ X . Different histories generally give rise to differentlocal unstable manifolds in X . In our situation, the disks subordinate to T − play the role ofunstable disks (Proposition 7.1 and the remark thereafter). Definition 7.5.
We say that the unstable manifolds of f fully depend on histories if theassignment ˆ x ( x , D (ˆ x )) is 1-1 on a set of full measure; in other words, if the intermediateprojection ˆ X → ˇ X is a measurable isomorphism.We say that the unstable manifolds do not depend on histories if ˇ π is a measurable isomor-phism but π : ˆ X → X is not (i.e. f is not essentially invertible). Przytycki proved in [Pr] that for C -generic Anosov endomorphisms of tori, the unstablemanifolds depend on histories. Mihailescu and Urbanski [MU] have proved the dependenceon histories for certain generic perturbations of saddle sets for holomorphic endomorphisms of P . It is natural to expect that for generic mappings with small topological degree (and notessentially invertible, see Example 3.2), the unstable manifolds (fully) depend on histories.However we do not know how to verify this, even on a single example! ⋄ We return to establishing the ergodic properties of µ . In the next theorem we analyze theconditionals induced by ˆ µ on the partition by local unstable manifolds in ˆ X . Theorem 7.6.
The conditional measures of ˆ µ along the unstable partition in ˆ X are inducedby the current T + . The unstable conditionals are well understood in ˇ X , so we will take advantage of the factthat ˆ X is the natural extension of ˇ X : if ˆ p ∈ ˆ X is a generic point, it has a local unstablemanifold W u loc (ˆ p ), and π is a homeomorphism from W u loc (ˆ p ) to a disk D through π ( p ). Thestatement of the theorem is that ˆ µ W u loc (ˆ p ) = c ( π − ) ∗ ( T + ˙ ∧ D ) ( c is a normalization constant).Notice here that it is important to consider the natural extension as a topological, and notonly measurable, object. To address topological issues in the natural extension, we use theordinary model of the shift acting on the space of histories, which is naturally a compactmetric space. Note also that if the unstable manifolds fully depend on histories, then theresult is obvious since ˆ X ≃ ˇ X . Proof.
We view ˆ X as the natural extension of ˇ X . So for notational ease we let π denote thenatural mapping ˆ X → ˇ X . Consider also the sequence of projections ( π n ) n ∈ Z with π n +1 =ˇ f ◦ π n .By construction, ˇ X admits a measurable partition into local unstable disks. Consider a flowbox P of positive measure in ˇ X , that is, a sub-lamination of one of the sets ˇ E k of Theorem6.1, made up of disks of size Q . We denote by p the generic point of P , and by ξ the naturalpartition of P by disks, so that ξ ( p ) is the disk through p .From the discussion following Definition 5.2, we know that every Q -distinguished historyof p ∈ P comes along with a sequence of inverse branches of ξ ( p ). Reducing the size of P ifnecessary, the set of Q -distinguished inverse branches of the disks of P forms a set of positivemeasure (a flow box) in ˆ X , that we will denote by ˆ P and which is naturally laminated byunstable manifolds. We denote by ˆ ξ u the partition of ˆ P into unstable disks. By transitivityof the conditional expectation, the conditionals induced by ˆ µ on the atoms ˆ ξ are also inducedby ˆ µ ˆ P = µ ( ˆ P ) ˆ µ | ˆ P . Since we can exhaust ˆ X up to a set of arbitrarily small measure with flowboxes, it will be enough to understand the conditionals on ˆ P .The partition ξ induces a (coarse) partition ˆ ξ = π − ( ξ ) on ˆ P , defined by ˆ ξ (ˆ p ) = π − ξ ( π (ˆ p )).Consider an atom D of the partition ξ on ˇ X and look at the part of ˇ f − n ( D ) corresponding EOMETRIC CURRENTS AND ERGODIC THEORY 29 to the branches belonging to ˆ P . This is a union of univalent inverse branches of ˇ f n , so itinherits a natural finite partition into inverse branches. We can reformulate this as follows:given an atom C ∈ ˆ ξ , π − n ( C ) admits a partition into finitely many pieces corresponding tothe inverse branches of ˇ f n along π ( C ). This induces a refinement of ˆ ξ that we denote by ˆ ξ − n (“separating inverse branches of order n ”, see figure 2). It is clear that ˆ ξ − n is an increasingsequence of partitions such that W ∞ n =0 ˆ ξ − n = ˆ ξ u , the partition of ˆ P into unstable leaves, upto a set of zero measure. 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(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) PSfrag replacements = ˆ ξ = ˆ ξ − = ˆ ξ − DD ffffffff ˆ Pπ Figure 2.
Construction of ˆ ξ u . Inverse branches of D in ˆ P are depicted onthe left, and the inductive construction of the partition on the rightBy Theorem 6.1, the conditionals of ˇ µ on the atoms of ξ are induced by T + , i.e. for a.e. p , ˇ µ ξ ( p ) = c ( p )( T + ˙ ∧ [ ξ ( p )]). Consider the conditionals induced by ˆ µ on ˆ ξ . Applying theprojection lemma 2.2 with ( ˜ Y , ν ) = ( ˆ X, ˆ µ ) and the projection π , we get that for almostevery atom C of ˆ ξ , ( π ) ∗ (ˆ µ C ) = ˇ µ π ( C ) .Consider now the disintegration of ˆ µ relative to the refined partition ˆ ξ − n . We will provethat for every n , if C is a generic atom of ˆ ξ − n , ( π ) ∗ ˆ µ C = ˇ µ π ( C ) . Let us first see why thisimplies the statement of the theorem. Recall that ˆ ξ − n increases to ˆ ξ u . Hence for ˆ µ a.e. ˆ p and every measurable function ψ , ˆ µ ˆ ξ − n (ˆ p ) ( ψ ) → ˆ µ ˆ ξ u (ˆ p ) ( ψ ) (this is the Martingale ConvergenceTheorem, see [PU]). Now if ψ is of the form ϕ ◦ π , the statement that ( π ) ∗ ˆ µ ˆ ξ − n (ˆ p ) = ˇ µ ξ ( π (ˆ p )) ,implies that for every n , ˆ µ ˆ ξ − n (ˆ p ) ( ψ ) = ˇ µ ξ ( π (ˆ p )) ( ϕ ) is independent of n . So we get thatˆ µ ˆ ξ (ˆ p ) ( ψ ) = ˇ µ ξ ( π (ˆ p )) ( ϕ ). In other words, ( π ) ∗ ˆ µ ˆ ξ u (ˆ p ) = ˇ µ ξ ( π (ˆ p )) . But now π is a measurableisomorphism π : ˆ ξ u (ˆ p ) → ξ ( π (ˆ p )) and ˇ µ ξ ( π (ˆ p )) is induced by T + , so the proof is finished. It remains to prove our claim that if C is a generic atom of ˆ ξ − n , ( π ) ∗ ˆ µ C = ˇ µ π ( C ) . Forthis, denote by D the disk D = π ( C ) and notice that C = ( π n ) − ( D − n ), where D − n isthe image of D by some inverse branch of f n . By Lemma 2.2 applied to π − n , we get that( π − n ) ∗ ˆ µ C = ˇ µ D − n . Since π = f n ◦ π − n , we thus obtain that ( π ) ∗ ˆ µ C = ( f n ) ∗ ˇ µ D − n . Nowwe know that the conditional ˇ µ D − n is induced by T + , and by the invariance property of T + we have that T + ∧ [ D ] = λ n ( f n ) ∗ ( T + ∧ [ D − n ]). After normalization, we conclude that( f n ) ∗ ˇ µ D − n = ˇ µ D and the result is proved. (cid:3) We can now compute the entropy, using the Rokhlin formula and the invariant “Pesinpartition” of [LS], as in [Du4]. See [BLS1] for a nice presentation of the material neededhere. The partition ξ is said to be f − -invariant if f − ξ is a refinement of ξ , i.e. for every x , f − ( ξ ( f ( x ))) ⊂ ξ ( x ). It is generating if W n ≥ f − n ξ is the partition into points; such partitionsallow the computation of entropy. Proposition 7.7 ([LS]) . There exists a measurable ˆ f − -invariant and generating partitionof ˆ X , whose atoms are open subsets of local unstable manifolds a.s.Proof. The proposition is stated in the context of Pesin’s theory applied to diffeomorphismsof manifolds in [LS], but it is well adapted to our situation. The exact requirements are listedin [LS, Prop. 3.3]. What is needed is a family of local manifolds V loc ( x ), and a set set Λ ℓ ofmeasure ≥ − ε ( ℓ ) such that- for x ∈ Λ ℓ , the manifolds V loc ( x ) have uniformly bounded geometry, and move con-tinuously;- ˆ f − n is uniformly exponentially contracting on V loc ( x ), x ∈ Λ ℓ .In our situation, we know that ˇ X can be written as a countable union of flow boxesas follows: first write ˇ X as a countable union of flow boxes P . Then write π − ( X ) as acountable union of flow boxes, up to a set of zero measure as follows: consider the increasingsequence of subdivisions Q i , subdivide P into smaller flow boxes of size Q i , and consider the Q i -distinguished histories of the smaller flow boxes.Now if P is a flow box of size Q and ˆ P is the set of its Q -distinguished histories, then ˆ P is naturally a flow box of ˆ X , where the plaques are unstable manifolds and the transversalis the set of Q -distinguished histories of a transversal of P . Moreover, the dynamics of ˆ f − n is uniformly exponentially contracting along the leaves. We leave the reader check that [LS,Prop. 3.1] can now be adapted easily –see also [QZ] for an adaptation of [LS] to a non-invertible situation, using Pesin’s theory. (cid:3) Corollary 7.8. h ( f, µ ) = log λ .Proof. The proof now follows from a classical argument, which we include for the reader’sconvenience. We compute the entropy in the natural extension. Let ˆ ξ u be the invariantpartition constructed above. Since ˆ ξ u is a generator and h ˆ µ ( ˆ f ) ≤ log λ is finite, h ˆ µ ( ˆ f ) equalsthe conditional entropy h ˆ µ ( ˆ f , ˆ ξ u ). Now the conditional entropy may computed by the Rokhlinformula: h ˆ µ ( ˆ f , ˆ ξ u ) = − Z log µ ˆ ξ u ( x ) ( ˆ f − ˆ ξ u ( x ))) d ˆ µ ( x ) = Z log J u ˆ µ ( x ) d ˆ µ ( x ) , where J u ˆ µ ( x ) := (cid:16) ˆ µ ˆ ξ u ( x ) (cid:0) ˆ f − ( ˆ ξ u ( ˆ f ( x ))) (cid:1)(cid:17) − is the unstable Jacobian. EOMETRIC CURRENTS AND ERGODIC THEORY 31
Since the ˆ ξ u ( x ) are open subsets of unstable manifolds in ˆ X , by Theorem 7.6 the condi-tionals ˆ µ ˆ ξ u ( x ) are induced by T + , that is, with the usual abuse of notation,ˆ µ ˆ ξ u ( x ) = T + ˙ ∧ [ ˆ ξ u ( x )] M (cid:0) T + ˙ ∧ [ ˆ ξ u ( x )] (cid:1) . From the invariance relation f ∗ T + = λ T + we deduce that T + ˙ ∧ [ ˆ f − ( ˆ ξ u ( ˆ f ( x )))] = (cid:0) T + ˙ ∧ [ ˆ ξ u ( x )] (cid:1)(cid:12)(cid:12) ˆ f − (ˆ ξ u ( ˆ f ( x ))) = 1 λ ˆ f ∗ ( T + ˙ ∧ [ ˆ ξ u ( ˆ f ( x ))]) , hence the unstable Jacobian J u ˆ µ satisfies the multiplicative cohomological equation J u ˆ µ ( x ) = λ ρ ( x ) ρ ( ˆ f ( x )) a.e. , where ρ ( x ) = M (cid:0) T + ˙ ∧ [ ˆ ξ u ( x )] (cid:1) . To prove that the integral of log J u ˆ µ equals log λ , we want to use Birkhoff’s Ergodic Theorem.The (additive) coboundary log ρ ( x ) − log ρ ( ˆ f ( x )) needn’t be integrable, nevertheless by theinvariance of the partition it is bounded from below (see [BLS1, Proposition 3.2] for moredetails). So Birkhoff’s Theorem applies and we conclude that h ˆ µ ( ˆ f ) = log λ . (cid:3) Product structure
We say that a measure has local product structure with respect to stable and unstablemanifolds if there is a covering by product subsets of positive measure, in which the vertical(resp. horizontal) fibers are exponentially contracted (resp. expanded) by the dynamics, andthe measure is isomorphic to a product measure in each of these subsets. This property isknown to have strong ergodic consequences, like the K and Bernoulli property (see [OW] formore details).
Theorem 8.1.
The measure ˆ µ has local product structure with respect to local stable andunstable manifolds in ˆ X .Proof. The proof is in two steps. We first show that ˇ µ has local product structure. This isactually a statement about the geometry of the currents T ± . Next we pass to the naturalextension. What is delicate in this step is to analyze the stable conditionals. Observe alsothat when unstable manifolds fully depend on histories, the second step is automatic. Step 1.
We begin by identifying product subsets in ˇ X . We let ˇ L u denote a “flow box for T − ”, that is, a sublamination of a flow box in ˇ X . We may write ˇ L u = S D u ∈ τ u D u , where D u are local unstable disks and τ u is an (abstract) transversal. Now suppose that D s is a disksubordinate to T + in X and that D s meets some disk in ˇ L u transversely in a single point.By definition of subordinate disks, there is a flow box L s = S D s ∈ τ s D s of positive mass for T + made of disks D s close to D s . Restricting ˇ L u if necessary, we may assume that each diskin L s intersects each disk in ˇ L u transversely in a single point.We lift leaves D s ∈ τ s to ˇ L u by setting ˇ D s := ˇ π − ( D s ) | ˇ L u and ˇ L s = S D s ∈ ˇ τ s ˇ D s . It is clearthat ˇ L u ∩ ˇ L s is a product set, homeomorphic to τ s × τ u . For later convenience we identify theabstract transversal τ s with a subset of some leaf D u ∈ τ u and likewise τ u with a subset of This might look quite cumbersome at first sight, but be careful that by definition the natural lifts ofdisks to the tautological extension never intersect! Here we play with the distinction between total and propertransform: we intersect the proper transform under ˇ π of D u with the total transform of D s . We put a checkon ˇ D s and not on D u to emphasize the fact that these two objects do not have the same status. of some ˇ D s . Since T + and T − intersect geometrically, there is a countable family of disjointproduct sets τ s × τ u as above, whose union has full ˇ µ measure.By the analytic continuation theorem for T + [Du3, Theorem 1.1] we know that T + | L s isuniformly laminar. Similarly, Theorem 1.6 implies that T − | ˇ π ( ˇ L u ) is uniformly woven and canhence be lifted as a uniformly laminar current on the “abstract” lamination ˇ L u . Abusingnotation, we denote this lift by T − | ˇ L u . Taking tautological extensions gives rise to a naturalproduct measure on ˇ L u ∩ ˇ L s , which, abusing notation again, we denote by T + | ˇ L s ˙ ∧ T − | ˇ L u . Toconclude that ˇ µ has local product structure, it remains to prove that this product measurecoincides with ˇ µ | ˇ L u ∩ ˇ L s .Recall from Theorem 6.1 that we have constructed ˇ µ as the increasing limit of the tauto-logical extensions of T + ˙ ∧ T −Q i . The measure T + ˙ ∧ T −Q i is in turn defined as an increasing limitof T + Q ′ ˙ ∧ T −Q i . Therefore, the restriction ˇ µ | ˇ L u ∩ ˇ L s is an increasing limit of restrictions of thetautological extensions ( T + Q ′ ˙ ∧ T −Q i ) | ˇ L u ∩ ˇ L s . By definition of the restriction of T + / − to L s/u , allthese measures are dominated by T + | ˇ L s ˙ ∧ T − | ˇ L u . So we infer that ˇ µ | ˇ L u ∩ ˇ L s ≤ T + | ˇ L s ˙ ∧ T − | ˇ L u .On the other hand T + | ˇ L s ˙ ∧ T − | ˇ L u is a measure supported on ˇ L u ∩ ˇ L s and dominated by ˇ µ sothe converse inequality is obvious. Step 1, reinforced.
To pass to the natural extension we actually need a stronger versionof the product structure of ˇ µ , where stable and unstable pieces are allowed to intersect manytimes. Consider a flow box ˇ L u as above, endowed with the abstract uniformly laminar cur-rent T − | ˇ L u , or equivalently, with an invariant transverse measure µ − . Define a measurabletransversal as a measurable set in ˇ L u intersecting each unstable leaf in ˇ L u along a discreteset. The transverse measure induces a positive measure µ − τ on each measurable transversal τ , invariant under the equivalence relation defined by the unstable leaves (see [MS, p.102]).More precisely, if τ and τ are two measurable transversals and φ : τ → τ is a measurableisomorphism preserving the leaves, then φ ∗ µ − τ = µ − τ .Using this formalism and the geometric intersection of the currents, we have a finer un-derstanding of the stable conditionals. Consider a set of positive measure, endowed with apartition ξ s , such that each piece ξ s is contained a countable union of disks subordinate to T + .We can thus consider the trace ˆ ξ s of this partition on ˇ L u , as done before. We further assumethat each atom of ˇ ξ s intersects the leaves of ˇ L u along a discrete set. So it is a measurabletransversal to ˇ L u . The conclusion is that the conditionals of ˇ µ (or equivalently, ˇ µ ˇ L u ) on thepieces ˇ ξ s are induced by the transverse measure µ − , i.e. ˇ µ ˇ ξ s = cµ − ˇ ξ s , with c a normalizationconstant. Step 2.
As above we view ˆ X as the natural extension of ˇ X , and denote by ( π n ) n ∈ Z thenatural projections ˆ X → ˇ X . Consider a product set P ≃ τ s × τ u as defined above. Forfine enough Q , the set of Q -distinguished histories of points in P has positive ˆ µ measure.Reducing P if necessary, we can assume that P is contained in a flow box of size Q , anddenote by ˆ P the set of Q -distinguished histories with x ∈ P . We will first prove that ˆ P is aproduct set (see also Figure 1) and next that ˆ µ | ˆ P has product structure.Denote by ˆ τ u the set of Q -distinguished histories of points in τ u . Recall that every Q -distinguished history of p ∈ P comes along with a sequence of exponentially contractinginverse branches defined on the disk of size Q on which p sits. So for every ˆ p ∈ ˆ τ u there existsa lift ˆ D u (ˆ p ) of D u ( π (ˆ p )) such that π : ˆ D u (ˆ p ) → D u ( π (ˆ p )) is an isomorphism. Clearly,ˆ P ⊂ S ˆ p ∈ ˆ τ u ˆ D u (ˆ p ). EOMETRIC CURRENTS AND ERGODIC THEORY 33
On the other hand, for every ˆ p ∈ ˆ P , the local stable manifold of ˆ p is the full preimage under π of the local stable manifold of π ( p ). More precisely, let ˆ D s (ˆ p ) = π − ( ˇ D s ( π (ˆ p ))) ∩ ˆ P . Thedynamics are exponentially contracting along the pieces ˆ D s . The set of pieces is parameterizedby τ s , or equivalently by the lift of τ s to some unstable disk ˆ D u (recall that τ s is identifiedwith a subset of some D u ). Since π is injective on ˆ D u (ˆ p ) and π ( ˆ D u (ˆ p ) ∩ ˆ D s (ˆ p )) ⊂ { π (ˆ p ) } ,we infer that ˆ D u (ˆ p ) ∩ ˆ D s (ˆ p ) = { ˆ p } . We conclude that ˆ P ≃ τ s × ˆ τ u is a product set.We now show that ˆ µ is a product measure in ˆ P . Since ˆ P is a product, we have twopartitions ˆ D s and ˆ D u , with a natural holonomy map between stable (resp. unstable) pieces.We have to prove that the conditionals induced by ˆ µ on the stable (resp. unstable) pieces areinvariant under holonomy.This is easier for unstable conditionals since we know them explicitly. Indeed, let ˆ D ui , i = 1 , h : ˆ D u → ˆ D u be the holonomy map. Define the correspondingobjects D ui and h in P by projecting under π . Recall that π is an isomorphism ˆ D ui → D ui .We have proved in Theorem 7.6 that ( π ) ∗ ˆ µ ˆ D ui = ˇ µ D ui . In addition, by the product structureof ˇ µ in P , we know that h ∗ ˇ µ D u = ˇ µ D u . Since h ◦ π = π ◦ ˆ h , we conclude that ˆ h ∗ ˆ µ ˆ D u = ˆ µ ˆ D u .Due to possible asymmetry between preimages, we cannot give an explicit description ofthe conditionals of ˆ µ on the stable partition ˆ D s . We nevertheless have enough informationto prove holonomy invariance. That is, if ˆ h : D s → D s is the holonomy map between twostable pieces, we will show that ˆ h ∗ ˆ µ ˆ D s = ˆ µ ˆ D s . Consider the measurable set P − n := π − n ( ˆ P ).We will use the following principle “the conditionals ˆ µ ˆ D s are completely determined by theconditionals induced by ˇ µ on ˇ f − n ˇ D s ∩ P − n , and the latter are invariant under holonomy”.To make the argument more accessible, we first make the following simplifying hypotheses:(i) ˆ P = π − − k ( P − k ) for some k ;(ii) for every unstable piece ˆ D u of the unstable partition of ˆ P , π − n ( ˆ D u ) is contained in asingle flow box of ˇ X .Restricting our attention to n ≥ k , we observe that assumption (i) implies ˆ P = π − − n ( P − n ).From this and the defining properties of ˆ µ we get that ( π − n ) ∗ ˆ µ ˆ P = ˇ µ P − n . Let ˇ D s, − n denote thepullback partition ˇ f − n ˇ D s of P − n . Since π = f n ◦ π − n we infer that for every p ∈ P − n , we have π − ˇ D s ( f n ( p )) ∩ ˆ P = π − − n ( ˇ f − n ˇ D s ( p ) ∩ P − n ). Or rather, in our notation, ˆ D s = π − − n ( ˇ D s, − n ).Passing to the conditionals, by Lemma 2.2 we infer that for a.e. ˆ p (with p − n = π − n (ˆ p )(16) ( π − n ) ∗ ˆ µ ˆ D s (ˆ p ) = ˇ µ ˇ D s, − n ( p − n ) . We define the holonomy map (depending on n ) between pieces of ˇ D s, − n naturally as follows:let ˆ D s and ˆ D s be two stable pieces and ˆ h be the holonomy map between them. For p − n = π − n (ˆ p ) ∈ ˇ D s, − n , let h ( p − n ) = π − n (ˆ h (ˆ p )) ∈ ˇ D s, − n (which is independent of the choice of ˆ p mapping to p − n ). By (ii), the points p − n and h ( p − n ) belong to the same flow box of ˇ X , andof course correspond under holonomy in this flow box.Let us prove that the conditionals ˇ µ ˇ D s, − n are invariant under h . It is enough to restrictˇ D s, − n to some flow box ˇ L u . To simplify notation, we continue to denote the restriction byˇ D s, − n . As the trace in L u of a holomorphic disk, ˇ D s, − n intersects unstable disks along discretesets (the proof of Proposition 7.1 shows that no open subset of D s, − n can be contained in anunstable leaf). Hence we are in position to apply the reinforced version of Step 1: the condi-tionals ˇ µ ˇ D s, − n are induced by the transverse measure associated to T − | ˇ L u , and by assumption (ii), intersection points do not escape ˇ L u by flowing under h ; hence h defines a measurableisomorphism ˇ D s, − n → ˇ D s, − n , respecting the leaves. It follows that the conditionals ˇ µ ˇ D s, − n are invariant under h .From (16) and this discussion, we deduce for every n and almost every pair of atoms ˆ D s ,ˆ D s , that ( π − n ) ∗ ˆ h ∗ ˆ µ ˆ D s = ( π − n ) ∗ ˆ µ ˆ D s , where ˆ h is the holonomy map ˆ D s → ˆ D s . It follows thatˆ h ∗ ˆ µ ˆ D s = ˆ µ ˆ D s . Indeed we have two measures ˆ h ∗ ˆ µ ˆ D s and ˆ µ ˆ D s on ˆ D s , agreeing on the σ -algebra F n generated by sets of the form π − − n ( A ). For every A ⊂ ˆ D s , we have A = T n ≥ k π − − n ( π − n ( A )).Hence the smallest σ -algebra containing all F n , n ≥ k , is the Borel σ -algebra. The assertionnow follows from standard measure theory.What remains now is to remove the simplifying assumptions. We will show that thesimplifying assumptions are true “up to subsets of small measure”. The details are a bitintricate; we start with a simple observation. Lemma 8.2.
Let ( A, ν ) be a probability space with a measurable partition ξ . Assume that B n is a sequence of sets with ν ( B n ) → . Then ν ξ ( p ) ( B n ∩ ξ ( p )) = 1 − ε ( p, n ) , with ε ( · , n ) → inprobability as n → ∞ . In particular there is a subsequence such that ε ( · , n ) → a.e. Consider the product set ˆ P as above, endowed with the partition ˆ D s . Since we do not nec-essarily have that ( π − n ) ∗ µ ˆ P = µ P n , we consider the sequence of sets π − − n ( P n ) = π − − n ( π − n ( P ))decreasing to ˆ P . We let E sn denote the partition π − ( ˇ D s ) of π − − n ( P n ). For every p ∈ ˆ P ,the sequence E sn (ˆ p ) decreases to ˆ D s ( p ), so ˆ µ π − − n ( P n ) | ˆ P is proportional to µ ˆ P . In terms of theconditionals induced on π − ( ˇ D s ), this implies that(17) µ E sn (ˆ p ) | ˆ D s ( p ) = u (ˆ p, n ) µ ˆ D s (ˆ p ) where u ( · , n ) is constant on ˆ D s ( p ) and increases to 1 a.e. With notation as before, observethat the analogue of (16) is now(18) ( π − n ) ∗ ˆ µ E sn (ˆ p ) = ˇ µ ˇ D s, − n ( p − n ) . We face two problems regarding holonomy. First, ˆ h is not defined everywhere on π − − n P n .Second, the holonomy h is not defined everywhere in P − n because points can escape flowboxes. Let ˆ R be the set (depending on n ) of points ˆ p ∈ ˆ P such that π − n ( ˆ D u (ˆ p )) is containedin a single flow box of ˇ X . By construction, this is a product set. Furthermore, the diametersof the disks π − n ( ˆ D u (ˆ p )) are bounded above by Cnλ − n/ , with C uniform in ˆ P . Thus as soonas π − n (ˆ p ) has distance at least Cnλ − n/ from the boundary of a flow box (in the leafwisedirection), we have D u (ˆ p ) ⊂ ˆ R . Since ˇ µ concentrates no mass on the boundary, we concludethat the relative measure of ˆ R in ˆ P tends to 1 as n → ∞ . If R − n = π − n ˆ R , then the holonomymap h along the leaves of ˇ X is well defined on R − n and preserves the conditionals induced by µ on the induced partition D s, − n ∩ R − n . The relationships among the sets we have introducedare summed up as follows: π − − n P n ⊃ ˆ P ⊃ ˆ R ⊂ π − − n R n ⊂ π − − n P n ;Moreover, ˆ h is well-defined on ˆ P , h is well-defined on R n , and π − n ◦ ˆ h = h ◦ π − n on ˆ P ∩ π − − n R n .Using (18) we deduce for a.e. ˆ p , ˆ p that(19) h ∗ (cid:2)(cid:0) ( π − n ) ∗ ˆ µ E sn (ˆ p ) (cid:1) | R − n (cid:3) = (cid:0) ( π − n ) ∗ ˆ µ E sn (ˆ p ) (cid:1) | R − n . EOMETRIC CURRENTS AND ERGODIC THEORY 35
Restricting measures does not commute with π − n but gives at least an inequality. Com-bining it with (17), we find (cid:0) ( π − n ) ∗ ˆ µ E sn (ˆ p ) (cid:1) | R − n ≥ ( π − n ) ∗ (cid:16) ˆ µ E sn (ˆ p ) | ˆ P ∩ π − − n R n (cid:17) = u (ˆ p , n )( π − n ) ∗ (cid:16) ˆ µ ˆ D s (ˆ p ) | π − − n R n (cid:17) . The left side of the inequality is a measure of at most unit mass, whereas by Lemma 8.2 themass of the right side is of the form 1 − ε (ˆ p , n ) (here ε ( · , n ) denotes a sequence of functionsconverging in probability to zero, possibly changing from line to line). So the right and leftsides differ by a measure of mass of at most ε (ˆ p , n ). By Lemma 8.2 again, ˆ µ ˆ D s (ˆ p ) | π − − n R n isclose to ˆ µ ˆ D s (ˆ p ) in mass. Therefore, finally we see that(20) M (cid:16)(cid:0) ( π − n ) ∗ ˆ µ E sn (ˆ p ) (cid:1) | R − n − ( π − n ) ∗ ˆ µ ˆ D s (ˆ p ) (cid:17) = ε (ˆ p , n ) . Since π − n ◦ ˆ h = h ◦ π − n on ˆ P ∩ π − − n R n the left side of (19) similarly satisfies h ∗ (cid:2)(cid:0) ( π − n ) ∗ ˆ µ E sn (ˆ p ) (cid:1) | R − n (cid:3) ≥ u (ˆ p , n )( π − n ) ∗ ˆ h ∗ (cid:16) ˆ µ ˆ D s (ˆ p ) | π − − n R n (cid:17) . Applying the same reasoning on masses yields(21) M (cid:16) h ∗ (cid:2)(cid:0) ( π − n ) ∗ ˆ µ E sn (ˆ p ) (cid:1) | R − n (cid:3) − ( π − n ) ∗ ˆ h ∗ ˆ µ ˆ D s (ˆ p ) (cid:17) = ε (ˆ p , n ) . From (20) and (21) we conclude that M (cid:16) ( π − n ) ∗ ˆ h ∗ ˆ µ ˆ D s (ˆ p ) − ( π − n ) ∗ ˆ µ ˆ D s (ˆ p ) (cid:17) = ε (ˆ p , n ) + ε (ˆ p , n ) . Consider as before the σ -algebra F n generated by π n . Given A in F n , we get that ˆ h ∗ ˆ µ ˆ D s (ˆ p ) ( A )and ( π − n ) ∗ ˆ µ ˆ D s (ˆ p ) ( A ) differ by at most ε (ˆ p , n ) + ε (ˆ p , n ). But since A ∈ F m for all m ≥ n ,we may pass to a subsequence and arrange for a.e. ˆ p , ˆ p that the difference tends to zero.Hence ˆ h ∗ ˆ µ ˆ D s (ˆ p ) = ˆ µ ˆ D s (ˆ p ) , which is what we wanted to prove. (cid:3) Saddle points
In this section we prove Theorem C. The proof is more classical and follows [BLS2] closely.
Step 1: Pesin theory.
As explained in the introduction, the proof relies on Pesin’s theory ofnon-uniformly hyperbolic dynamical systems. The applicability of this theory in our contextonly requires the assumption that p log d ( p, I + ∪ C f ) ∈ L ( µ ), as is neatly shown in [dT2].By using the Osedelets theorem or the foregoing study, we have a set of full ˆ µ measureprovided with an invariant splitting of the tangent space T p X = E u (ˆ p ) ⊕ E s ( p ) into an(exponentially) expanding direction and a contracting direction which depend measurablyon ˆ p . As indicated by the notation, E s ( p ) depends only on p = π (ˆ p ). By Pesin Theory,there exists an invariant set ˆ R ⊂ ˆ X of full ˆ µ measure such that for each ˆ p ∈ ˆ R , thereexists Lyapunov chart L (ˆ p ), which is a topological bidisk in X (in the terminology of [BLS2])centered at p = π (ˆ p ). The bidisk can be chosen to be the image, under the Riemannianexponential map, of an affine bidisk with axes E u (ˆ p ), E s ( p ) and measurably varying size r (ˆ p ). We can further assume that L (ˆ p ) does not intersect I + ∪ C f or f ( I + ∪ C f ) and that f | L (ˆ p ) is injective.The Lyapunov charts have the fundamental property that f : L (ˆ p ) → L ( ˆ f ˆ p ) defines aH´enon-like map of degree 1 (we use the terminology of [Du1]). That is, the cut-off image ofa graph over the horizontal (i.e. unstable) direction, is a graph. This property is referred to in [BLS2] as the “ u -overflowing property” of Lyapunov charts. The branches of f − have theoverflowing property in the vertical (i.e. stable) direction.We can also consider the sets L sn (ˆ p ) := n y ∈ L (ˆ p ) , ∀ ≤ j ≤ n, f j ( y ) ∈ L ( ˆ f j ˆ p ) o and L un (ˆ p ) := f n L sn ( ˆ f − n ˆ p ) , which converge exponentially fast to the local stable manifold W s loc ( p ) and unstable manifold W u loc (ˆ p ), respectively. Note that depending on the context, we will sometimes regard localstable and unstable manifolds as subsets of X and sometimes as subsets of ˆ X .We have shown in the proof of Theorem B i. that disks subordinate to T + are µ -a.e. expo-nentially contracted by f , while disks subordinate to T − are ˆ µ -a.e. exponentially contractedby distinguished preimages of f . On the other hand, the Pesin stable and unstable man-ifolds are unique. Therefore, Pesin stable and unstable manifolds coincide a.e. with diskssubordinate to T + and T − .If ˆ p and ˆ q are sufficiently close in ˆ X , then W s loc ( p ) ∩ W u loc (ˆ p ) is a single point classicallydenoted by [ˆ p, ˆ q ]. A subset is said to have product structure if it is closed under the operation[ · , · ]. A Pesin box is a compact, positive measure subset of ˆ X with product structure and apositive lower bound on the size of the associated Lyapunov charts. Step 2: constructing saddle points.
The basic step in the argument is the following: if g is a H´enon-like map of degree 1 in some topological bidisk B , then T k ∈ Z g k ( B ) is a singlesaddle fixed point q of g . Similarly T k ≥ g k ( B ) = W uB ( p ) and T k ≤ g k ( B ) = W sB ( p ). Here weemploy truncated iteration, in which points are omitted once they leave B .Fix ε >
0. There exists a compact set ˆ R ε with ˆ µ (cid:0) ˆ R ε (cid:1) > − ε , and where all constantsappearing above, as well as the stable and unstable directions and manifolds, vary continu-ously. As argued in [BLS2, Lemma 1] (stated in the context of polynomial automorphisms,but the proof extends without change to our situation), given η >
0, there exist finitely manyPesin boxes, each of diameter smaller than η , covering ˆ R ε . Let ˆ P be one of these Pesin boxes.Since the stable and unstable directions of points belonging to ˆ P are almost parallel, if η issufficiently small, there exists a “common Lyapunov chart” B , which is a topological bidisksuch that π ( ˆ P ) ⊂ B ⊂ \ ˆ p ∈ ˆ P L (ˆ p ) . Now if n is large enough and ˆ p ∈ ˆ P ∩ ˆ f − n ( ˆ P ) (of course there are infinitely many such n ), f n induces a H´enon-like map of degree 1 in B that sends p = π (ˆ p ) to f n ( p ). We infer that f n has a saddle fixed point q exponentially (in n ) close to π ([ˆ p, ˆ f n (ˆ p )]) ⊂ π ( ˆ P ).Now let ˆ q be the unique periodic point in ˆ X projecting to q . We claim that ˆ q is close to ˆ P .By construction, for k ∈ N and 0 ≤ i ≤ n − f i + kn ( q ) ∈ L ( ˆ f i ˆ p ). Since ˆ q is periodic we inferthat for k ∈ Z , π ˆ f i + kn (ˆ q ) ∈ L ( ˆ f i ˆ p ). In particular if i is a fixed integer we can arrange sothat for | i | ≤ i , π ˆ f − i (ˆ q ) is close to π ˆ f − i ( ˆ P ). Thus ˆ q is close to ˆ P relative to the productmetric on ˆ X , as claimed.At this stage we know that saddle periodic points accumulate everywhere on Supp( µ ) andSupp(ˆ µ ). Step 3: equidistribution.
Given ˆ p ∈ ˆ P ∩ ˆ f − n ( ˆ P ) as in Step 2, let B sn ( p ) be the connectedcomponent of B ∩ f − n B containing p . Let us first show that B sn ( p ) ⊂ L sn (ˆ p ). For this, suppose EOMETRIC CURRENTS AND ERGODIC THEORY 37 that there exists p ′ ∈ B sn ( p ) and a smallest integer i ≥ f i ( p ′ ) / ∈ L ( ˆ f i (ˆ p )). Let γ be a path joining p and p ′ ; f i ( γ ) is not contained in L ( ˆ f i (ˆ p )), so it must intersect the verticalboundary of that polydisk. Hence by the H´enon-like property, for all subsequent iterates j ≥ i , f j ( γ ) intersects the vertical boundary of L ( ˆ f j (ˆ p )), which contradicts the fact that f n ( B sn ( p )) ⊂ B .Since B sn ( p ) ⊂ L sn (ˆ p ), the behavior of f n on B sn ( p ) is that of n successive iterates of aH´enon-like map of degree 1. In particular we infer that B sn ( p ) is a vertical sub-bidisk of B ,while f n ( B sn ( p )) is horizontal. Also, exactly as in [BLS2, Lemma 3] we get that(22) W sloc ( p ) ∩ B ⊂ B sn ( p ) and W uloc (ˆ p ) ∩ B sn ( p ) ⊂ (cid:0) f n | L sn (ˆ p ) (cid:1) − (cid:16) W uloc ( ˆ f n ˆ p ) ∩ B (cid:17) . The sets π − ( B sn ) induce a partition of ˆ P ∩ ˆ f − n ( ˆ P ). If T is an atom of this partition, itis clear that the construction of Step 2 associates to all points ˆ p ∈ T the same saddle point q = q ( T ), and that the mapping T q ( T ) is injective. From (22) and the fact that the localstable manifold of ˆ p in ˆ X is the full preimage under π of W s loc ( p ) ⊂ X , we get that T hasproduct structure. That is, if ˆ p , ˆ p ∈ T , then [ˆ p , ˆ p ] ∈ T .Now since ˆ µ has product structure and its unstable conditionals are induced by the current T + (Theorems 7.6 and 8.1), we can reproduce [BLS2, Lemma 5] and conclude that for anyatom T of the partition, ˆ µ ( T ) ≤ λ − n ˆ µ ( ˆ P ).Let SFix n be the set of saddle periodic points of period (dividing) n in ˆ X , and ˆ ν n = λ − n P q ∈ SFix n δ q . If ˆ P δ denotes the δ -neighborhood of ˆ P , we obtain that for large n ,(23) λ − n ˆ µ ( ˆ P ) n SFix n ∩ ˆ P δ o = ˆ µ ( ˆ P )ˆ ν n ( ˆ P δ ) ≥ X ˆ µ ( T ) = ˆ µ ( ˆ P ∩ ˆ f − n ˆ P ) −→ ˆ µ ( ˆ P ) , whence lim inf ˆ ν n ( ˆ P δ ) ≥ ˆ µ ( ˆ P ). If ˆ ν is any cluster point of the sequence ˆ ν n , we conclude thatfor any δ >
0, ˆ ν ( ˆ P δ ) ≥ ˆ µ ( ˆ P ), thus ˆ ν ( ˆ P ) ≥ ˆ µ ( ˆ P ). Since any open subset in ˆ X can be coveredup to a set of small ˆ µ measure by disjoint Pesin boxes, we conclude that ˆ ν ≥ ˆ µ Now assume (see Step 5 below) that we have an estimate n ≤ λ n + o ( λ n ). Thenlim sup ν n ( X ) ≤ ≤ µ ( X ). From this and the previous paragraph we conclude that µ =lim ν n .In the (unexpected?) event that the estimate n ≤ λ n + o ( λ n ) fails, then we cancertainly replace SFix n with a subset P n ⊂ SFix n of size P n ∼ λ n for which (23) remainsvalid relative to a countable collection of Pesin boxes sufficient to provide disjoint “nearcovers” of any open set. Hence the measures ν n defined using P n instead of SFix n will againconverge to µ . Step 4: saddle points in
Supp( µ ) . We now prove that the saddle points constructed in Step2 lie inside Supp( µ ). Here the argument is identical to [Du4], we sketch it for completeness(see also [BLS1]).Given a Pesin box ˆ P as above, let L s ⊂ X be the local stable lamination of π ( ˆ P ), thatis, the union of local stable manifolds. Likewise we can define the local unstable web L u = S ˆ p ∈ ˆ P W u loc (ˆ p ). Let S + = T + | L s and S − = T − | L u . These currents are uniformly geometric anddominated by T + and T − , respectively. Furthermore, from our understanding of µ we knowthat S + > S − >
0. With notation as in Step 2, let now g be the H´enon-like map in B induced by f n . Therefore, λ − nk ( g k ) ∗ S + is a non-trivial, uniformly laminar current dominatedby T + . As n → ∞ its support converges in the Hausdorff sense to the local stable manifoldof q , where q = T n ∈ Z g k ( B ) is the saddle point that we have just constructed. Similarly, we have corresponding currents 0 < λ kn ( g k ) ∗ S − ≤ T − with supports converging to the localunstable manifold of (the periodic history of) ˆ q . Hence we obtain a sequence of measures0 < µ k = 1 λ kn ( g k ) ∗ S + ∧ λ kn ( g k ) ∗ S − ≤ µ with supports converging to q . We conclude that q ∈ Supp( µ ). Step 5: counting periodic points.
The second statement in Theorem C requires a boundof the form
P er n ≤ λ n + o ( λ n ) on the number of isolated periodic points. When f hasno curves of periodic points, this is classical. The Lefschetz Fixed Point Formula [Fu, p.314]asserts that { ∆ } · n Graph( f k ) o = X ≤ p,q ≤ ( − p + q Trace (cid:16) ( f k ) ∗ | H p,q ( X, R ) (cid:17) , where ∆ is the diagonal in X × X . If f k has no curve of fixed points, the intersection productis the sum of the multiplicities of periodic points plus a (nonnegative) term coming from theindeterminacy set (see e.g. [DF, § ∩ Graph( f k ) have dimension 0,hence give positive contribution to the intersection product. In particular the left hand sideof this inequality dominates the number of fixed points of f k .On the other hand, the dominating term on the right hand side is given by the trace of theaction on H , , which is λ n + o ( λ n ). Indeed, observe first that by the small topological degreeassumption, the action on H predominates. Now, when X is rational, dim H , = dim H , =0 so we are done. When X is irrational it can be checked directly that (cid:13)(cid:13) ( f k ) ∗ | H , (cid:13)(cid:13) . λ n/ (ageneral argument for this is given in [Di, Proposition 5.8]), and we are also done in this case.When f admits curves of periodic points and X = P or X = P × P , the result followsfrom a slightly more sophisticated argument from Intersection Theory [Fu, § X × X = P × P or ( P × P ) . This manifold has theproperty that its tangent bundle is generated by its sections (see [Fu, Example 12.2.1]), inwhich case the contribution of every irreducible component of ∆ ∩ Graph( f k ) to the intersec-tion product { ∆ } · (cid:8) Graph( f k ) (cid:9) is nonnegative [Fu, Corollary 12.2] (see also [Fu, Example16.2.2]). As before we conclude that the number of isolated fixed points of f n is controlledby λ n + o ( λ n ).Observe that if X is any rational surface, by using a birational conjugacy between f anda (possibly non algebraically stable) map on P , this argument shows that the number ofisolated fixed points of f n is controlled by Cλ n + o ( λ n ) for some C . Remark . Under the same assumptions as in Theorem C we can also adapt [BLS2, Theorem2] and obtain that the Lyapunov exponents of µ can be evaluated by averaging on saddleorbits, which is an important fact in bifurcation theory. References [BeDi] Bedford, Eric; Diller, Jeffrey.
Energy and invariant measures for birational surface maps.
Duke Math.J. 128 (2005), no. 2, 331–368[BS] Bedford, Eric; Smillie, John.
Polynomial diffeomorphisms of C . III. Ergodicity, exponents and entropyof the equilibrium measure. Math. Ann. 294 (1992), 395-420.[BLS1] Bedford, Eric; Lyubich, Mikhail; Smillie, John.
Polynomial diffeomorphisms of C . IV. The measureof maximal entropy and laminar currents. Invent. Math. 112 (1993), 77-125.
EOMETRIC CURRENTS AND ERGODIC THEORY 39 [BLS2] Bedford, Eric; Lyubich, Mikhail; Smillie, John.
Distribution of periodic points of polynomial diffeo-morphisms of C . Invent. Math. 114 (1993), 277-288.[Br] Briend, Jean-Yves.
Propri´et´e de Bernoulli pour les extensions naturelles des endomorphismes de CP k . Ergodic Theory Dynamical Systems 21 (2001), 1001-1007.[BrDu] Briend, Jean-Yves; Duval, Julien.
Deux caract´erisations de la mesure d’´equilibre des endomorphismesde CP k . Publ. Math. Inst. Hautes ´Etudes Sci. No. 93 (2001), 145-159.[Ca] Cantat, Serge.
Dynamique des automorphismes des surfaces K3.
Acta Math. 187 (2001) 1-57.[dT1] De Th´elin, Henry.
Sur la construction de mesures selles.
Ann. Inst. Fourier, 56 (2006), 337-372.[dT2] De Th´elin, Henry.
Sur les exposants de Lyapounov des applications m´eromorphes.
Invent. Math. 172(2008), 89-116.[DV] De Th´elin, Henry; Vigny, Gabriel.
Entropy of meromorphic maps and dynamics of birational maps.
Preprint (2008). M´emoires de la SMF, to appear.[DDG1] Diller, Jeffrey; Dujardin, Romain; Guedj, Vincent.
Dynamics of meromorphic maps with small topo-logical degree I: from cohomology to currents.
Indiana Univ. Math. J., to appear.[DDG2] Diller, Jeffrey; Dujardin, Romain; Guedj, Vincent.
Dynamics of meromorphic maps with small topo-logical degree II: energy and invariant measure.
Comment. Math. Helvet., to appear.[DF] Diller, Jeffrey; Favre, Charles.
Dynamics of bimeromorphic maps of surfaces.
Amer. J. Math. 123(2001), no. 6, 1135–1169.[Di] Dinh, Tien Cuong.
Suites d’applications m´eromorphes multivalu´ees et courants laminaires.
J. Geom.Anal. 15 (2005), 207–227.[DS] Dinh, Tien Cuong; Sibony, Nessim.
Dynamique des applications d’allure polynomiale.
J. Math. PuresAppl. (9) 82 (2003), 367-423.[Du1] Dujardin, Romain.
H´enon-like mappings in C . Amer. J. Math. 126 (2004), 439-472.[Du2] Dujardin, Romain.
Sur l’intersection des courants laminaires.
Pub. Mat. 48 (2004), 107-125.[Du3] Dujardin, Romain.
Structure properties of laminar currents.
J. Geom. Anal. 15 (2005), 25-47.[Du4] Dujardin, Romain.
Laminar currents and birational dynamics.
Duke Math. J. 131 (2006), 219-247.[Duv] Duval, Julien.
Singularit´es des courants d’Ahlfors.
Ann. Sci. ´Ecole Norm. Sup. (4) 39 (2006), 527–533.[FJ] Favre, Charles; Jonsson, Mattias.
Dynamical compactifications of C . Preprint (2007). Annals ofMath., to appear.[FS] Fornaess, John Erik; Sibony, Nessim
Complex dynamics in higher dimension. II , Modern methods incomplex analysis, Ann. of Math. Studies 135–182. Princeton Univ. Press.[FLM] Freire, Alexandre; Lopes, Artur; Ma˜n´e, Ricardo
An invariant measure for rational maps.
Bol. Soc.Brasil. Mat. 14 (1983), 45–62.[Fu] Fulton, William,
Intersection theory.
Second edition. Ergebnisse der Mathematik und ihrer Grenzge-biete. 3. Folge. A Series of Modern Surveys in Mathematics. Springer-Verlag, Berlin, 1998.[Gh] Ghys, Etienne.
Laminations par surfaces de Riemann.
Dynamique et g´eom´etrie complexes (Lyon,1997), Panoramas et Synth`eses, 8, 1999.[Gr] Gromov, Mikhail.
On the entropy of holomorphic maps.
Enseign. Math. (2) 49 (2003), 217–235.[G1] Guedj, Vincent.
Dynamics of polynomial mappings of C . Amer. J. Math. (2002), no. 1, 75–106.[G2] Guedj, Vincent.
Ergodic properties of rational mappings with large topological degree.
Ann. of Math.161 (2005).[G3] Guedj, Vincent.
Entropie topologique des applications m´eromorphes.
Ergodic Theory Dynam. Systems25 (2005), no. 6, 1847–1855.[Led] Ledrappier, Fran¸cois.
Some properties of absolutely continuous invariant measures on an interval ,Ergodic Theory Dynam. Systems 1 (1981), 77–93.[LS] Ledrappier, Fran¸cois; Strelcyn, Jean-Marie.
A proof of the estimation from below in Pesin’s entropyformula.
Ergodic Theory Dynam. Systems 2 (1982), 203–219.[Lel] Lelong, Pierre.
Propri´et´es m´etriques des vari´et´es analytiques complexes d´efinies par une ´equation.
Ann. Sci. ´Ecole Norm. Sup. (3) 67, (1950). 393–419.[Ly] Lyubich, Mikhail
Entropy properties of rational endomorphisms of the Riemann sphere.
Ergodic The-ory Dynam. Systems 3 (1983), 351-385.[MU] Mihailescu, Eugen; Urba´nski, Mariusz.
Holomorphic maps for which the unstable manifolds dependon histories.
Discrete Contin. Dyn. Syst. 9 (2003), no. 2, 443–450.[MS] Moore, Calvin C.; Schochet, Claude.
Global analysis on foliated spaces.
Mathematical Sciences Re-search Institute Publications, 9. Springer-Verlag, New York, 1988. [OW] Ornstein, Donald; Weiss, Benjamin.
On the Bernoulli nature of systems with some hyperbolic structure.
Ergodic Theory Dynam. Systems 18 (1998), 441–456.[Pr] Przytycki, Feliks.
Anosov endomorphisms.
Studia Math. 58 (1976), 249–285.[PU] Przytycki, Feliks; Urba´nski, Mariusz.
Conformal fractals, Ergodic Theory methods.
Book to appear,available online at ∼ urbanski/book1.html [QZ] Qian, Min; Zhu, Shu. SRB measures and Pesin’s entropy formula for endomorphisms.
Trans. Amer.Math. Soc. 354 (2002), 1453–1471.[Ro] Rohlin, V. A.
Lectures on the entropy theory of transformations with invariant measure.
Uspehi Mat.Nauk 22 1967 no. 5 (137), 3–56. Transl. in Russian Math. Surveys 22 (1967), no. 5, 1–52.[Y] Yomdin, Yuri.
Volume growth and entropy.
Israel J. Math. 57 (1987), 285-300.
Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556
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