Path connectedness and entropy density of the space of ergodic hyperbolic measures
aa r X i v : . [ m a t h . D S ] J un PATH CONNECTEDNESS AND ENTROPY DENSITY OF THESPACE OF HYPERBOLIC ERGODIC MEASURES
ANTON GORODETSKI AND YAKOV PESIN
Abstract.
We show that the space of hyperbolic ergodic measures of a givenindex supported on an isolated homoclinic class is path connected and entropydense provided that any two hyperbolic periodic points in this class are ho-moclinically related. As a corollary we obtain that the closure of this space isalso path connected. Introduction
In this paper we consider homoclinic classes of periodic points for C α dif-feomorphisms of compact manifolds and we discuss two properties of the spaceof invariant measures supported on them and equipped with the weak ∗ -topology– connectedness and entropy density of the subspace of hyperbolic ergodic mea-sures. The study of connectedness of the latter space was initiated by Sigmund ina short article [32]. He established path connectedness of this space in the caseof transitive topological Markov shifts and as a corollary, of transitive Axiom A diffeomorphisms. Sigmund’s idea was to show first that any two periodic measures(i.e., invariant atomic measures on periodic points) can be connected by a contin-uous path of ergodic measures and second that if one of the two periodic measureslies in a small neighborhood of another one, then the whole path can be chosen tolie in this neighborhood. In order to carry out the first step Sigmund shows thatany periodic measure can be approximated by a Markov measure and that anytwo Markov measures can be connected by a path of Markov measures. We useSigmund’s idea in our proof of Theorem 1.1.A different approach to Sigmund’s theorem is to show that ergodic measures on atransitive topological Markov shift are dense in the space of all invariant measures.Since the latter space is a Choquet simplex and ergodic measures are its extremalpoints, it means that this space is the Poulsen simplex (which is unique up to anaffine homeomorphism). The desired result now follows from a complete descriptionof the Poulsen simplex given in [28] (see also [17]).Since Sigmund’s work the interest to the study of connectedness of the spaceof hyperbolic ergodic measures has somehow been lost, and only recently it hasregained attention. In [18] Gogolev and Tahzibi, motivated by their study of
Date : June 9, 2017.1991
Mathematics Subject Classification. At the time of writing this paper there is no single reference to the paper by Sigmund [32] in
MathSciNet . Soon after this paper was completed, several new works related to the subject appeared, see[9, 13]. existence of non-hyperbolic invariant measures, raised a question of whether thespace of ergodic measures invariant under some partially hyperbolic systems ispath connected. This includes, in particular, the famous example by Shub andWilkinson [31]. Some results on connectedness and other topological properties ofthe space of invariant measures were obtained in [8, 17].All known proofs of connectedness of the space of invariant measures are based onapproximating invariant measures by either measures supported on periodic orbitsor Markov ergodic measures supported on invariant horseshoes. It is therefore natu-ral to ask whether such approximations can be arranged to also ensure convergenceof entropies. If this is possible, the space of approximants is called entropy dense .Some results in this direction were obtained in [23]. We stress that approximating hyperbolic ergodic measures with positive entropy by “nice” measures supported oninvariant horseshoes so that the convergence of entropies is also guaranteed, wasfirst done by Katok in [25] (see also [2, 26]). We use this result in the proof of ourTheorem 1.5 where we approximate also some hyperbolic ergodic measures withzero entropy as well as non-ergodic measures.We shall now state our results. Consider a C α -diffeomorphism f : M → M ofa compact smooth manifold M . Let p ∈ M be a hyperbolic periodic point. By theindex s ( p ) of p we mean the dimension of the invariant stable manifold of p .We say that a hyperbolic periodic point q ∈ M is homoclinically related to p and write q ∼ p if the stable manifold of the orbit of q intersects transversely theunstable manifold of the orbit of p and vice versa. Notice that this is an equivalencerelation. We denote by H ( p ) the homoclinic class associated with the point p , thatis the closure of the set of hyperbolic periodic points homoclinically related to p .Note that H ( p ) is f -invariant. Homoclinic classes were introduced by Newhouse in[29].A basic hyperbolic set of an Axiom A diffeomorphism gives the simplest exampleof a homoclinic class, but in general the set H ( p ) can have a much more complicatedstructure and dynamical properties. In particular, it can contain non-hyperbolicperiodic points, and it can support non-hyperbolic (periodic or not) measures in arobust way, see [3, 5, 11, 27] for a more detailed discussion. It can also contain ina robust way hyperbolic periodic orbits whose index is different than the index of p , see [4, 21, 20]. Moreover – and this is of importance for us in this paper – theremay exist hyperbolic periodic points in H ( p ) of the same index as p that are not homoclinically related to p , see [11, 15]. Besides, it can happen that periodic orbitsoutside the homoclinic class H ( p ) accumulate to H ( p ); for example, this is partof the Newhouse phenomena, and also occurs in the family of standard maps, see[16, 19]. We wish to avoid both of these complications, and we therefore, imposethe following crucial requirements on the homoclinic class H ( p ):(H1) For any hyperbolic periodic point q ∈ H ( p ) with s ( q ) = s ( p ) we have q ∼ p . (H2) The homoclinic class H ( p ) is isolated, i.e., there is an open neighborhood U ( H ( p )) of H ( p ) such that H ( p ) = T n ∈ Z f n ( U ( H ( p ))) . The stable (respectively, unstable) manifold of the orbit of a periodic point is the unionof stable (respectively, unstable) manifolds through every point on the orbit. If q ∼ p , then s ( q ) = s ( p ). It is conjectured that existence of non-hyperbolic ergodic measures is a characteristic propertyof non-hyperbolic homoclinic classes, see [3, 11].
ATH CONNECTEDNESS AND ENTROPY DENSITY 3
We stress that these requirements do hold in many interesting cases, see examples inSection 2. In particular, Condition (H2) holds if the map f has only one homoclinicclass. This is the case in Examples 1 and 2 in Section 2. We also note a result in[8] that is somewhat related to Condition (H2): if the map f admits a dominatedsplitting of index s , then a linear combination of hyperbolic ergodic measures ofindex s can be approximated by a sequence of hyperbolic ergodic measures of index s if and only if their homoclinic classes coincide.The space of all invariant ergodic measures supported on H ( p ) can be extremelyrich and contain hyperbolic measures with different number of positive Lyapunovexponents as well as non-hyperbolic measures. We denote by M p the space of allhyperbolic invariant measures supported on H ( p ) for which the number of negativeLyapunov exponents at almost every point is exactly s ( p ). We say that µ has index s ( p ). Further, we denote by M ep the space of all hyperbolic ergodic measures in M p . We assume that the space M p is equipped with the weak ∗ -topology. Theorem 1.1.
Under Conditions (H1) and (H2) the space M ep is path connected. Notice that without Conditions (H1) and (H2) the conclusion of Theorem 1.1may fail, see Subsection 2.2.It follows immediately from Theorem 1.1 that the closure of M ep is connected.In fact, a stronger statement holds. Theorem 1.2.
Under Conditions (H1) and (H2) the closure of the space M ep ispath connected. Remark 1.3.
It is interesting to notice that the closure of M ep is not a Choquetsimplex (and hence, not a Poulsen simplex), see Proposition 2.7 in [9].We shall now discuss the entropy density of the space M ep . Definition 1.4.
A subset S ⊆ M p is entropy dense in M p if for any µ ∈ M p there exists a sequence of measures { ξ n } n ∈ N ⊂ S such that ξ n → µ and h ξ n → h µ as n → ∞ . Theorem 1.5.
Under Conditions (H1) and (H2) the space M ep is entropy densein M p . Examples
In this section we present some examples that illustrate importance of Conditions(H1) and (H2).2.1.
Non-hyperbolic homoclinic classes satisfying Conditions (H1) and(H2).
We describe a class of diffeomorphisms with a partially hyperbolic attrac-tor which is the homoclinic class of any of its periodic points and which satisfiesConditions (H1) and (H2). We follow [10]. Let f be a C α diffeomorphism ofa compact smooth manifold M and Λ a topological attractor for f . This meansthat there is an open set U ⊂ M such that f ( U ) ⊂ U and Λ = T n ≥ f n ( U ). Weassume that Λ is a partially hyperbolic set for f , that is for every x ∈ Λ there is aninvariant splitting of the tangent space T x M = E s ( x ) ⊕ E c ( x ) ⊕ E u ( x ) into stable E s ( x ), central E c ( x ) and unstable E u ( x ) subspaces such that with respect to someRiemannian metric on M we have that for some constants0 < λ < λ < λ < λ , λ < , λ > A. GORODETSKI AND YA. PESIN the following holds:(1) k df v k < λ k v k for every v ∈ E s ( x ),(2) λ k v k < k df v k < λ k v k for every v ∈ E c ( x ),(3) k df v k > λ k v k for every v ∈ E u ( x ).If Λ is a partially hyperbolic attractor for f , then for every x ∈ Λ we denoteby V u ( x ) and W u ( x ) the local and respectively global unstable leaves through x .It is known that for every x ∈ Λ and y ∈ W u ( x ) one has T y W u ( x ) = E u ( y ), f ( W u ( x )) = W u ( f ( x )) and W u ( x ) ⊂ Λ. Moreover, the collection of all globalunstable leaves W u ( x ) forms a continuous lamination of Λ with smooth leaves, andif Λ = M , then it is a continuous foliation of M with smooth leaves.An invariant measure µ on Λ is called a u -measure if the conditional measures itgenerates on local unstable leaves V u ( x ) are equivalent to the leaf volume on V u ( x )induced by the Riemannian metric. It is shown in [30] that any partially hyperbolicattractor admits a u -measure: any limit measure for the sequence of measures µ n = 1 n n − X k =0 f k ∗ m is a u -measure on Λ. Here m is the Riemannian volume in a sufficiently small neigh-borhood of the attractor (see [30] for more details and other ways for constructing u -measures).In general a u -measure may have some or all Lyapunov exponents along thecentral direction to be zero. Therefore, following [10] we say that a u -measure µ has negative central exponents on an invariant subset A ⊂ Λ of positive measure iffor every x ∈ A and v ∈ T x E c ( x ) the Lyapunov exponent χ ( x, v ) < f | Λ:(D) for every x ∈ Λ the positive semi-trajectory of the global unstable leaf W u ( x ) is dense in Λ, that is [ n ≥ f n ( W u ( x )) = [ n ≥ W u ( f n ( x )) = Λ . Condition (D) clearly holds if the unstable lamination is minimal, i.e., if every leafof the lamination is dense in Λ. It is shown in [10] that if µ is a u -measure on Λwith negative central exponents on an invariant subset of positive measure and if f satisfies Condition (D), then 1) µ has negative central exponents at almost everypoint x ∈ Λ; 2) µ is the unique u -measure for f supported on the whole Λ; and 3)the basin of attraction for µ coincides with the open set U .It is easy to see that in this case:(1) hyperbolic periodic points whose index is equal to the dimension of thestable leaves are dense in the attractor Λ; the homoclinic class of each ofthese periodic points coincides with Λ;(2) the homoclinic class satisfies Conditions (H1) and (H2), and hence, Theo-rems 1.1, 1.2, and 1.5 are applicable.Let f be a partially hyperbolic diffeomorphism which is either 1) a skew productwith the map in the base being a topologically transitive Anosov diffeomorphism Clearly, the Lyapunov exponents in the stable direction are negative while the Lyapunovexponents in the unstable direction are positive. This dimension is dim E s + dim E c . ATH CONNECTEDNESS AND ENTROPY DENSITY 5 or 2) the time-1 map of an Anosov flow. If f is a small perturbation of f then f is partially hyperbolic and by [24], the central distribution of f is integrable.Furthermore, the central leaves are compact in the first case and there are compactleaves in the second case. It is shown in [10] that f has minimal unstable foliationprovided there exists a compact periodic central leaf C (i.e., f ℓ ( C ) = C for some ℓ ≥
1) for which the restriction f ℓ |C is a minimal transformation.Furthermore, it follows from the results in [1] that starting from a volume pre-serving partially hyperbolic diffeomorphism f with one-dimensional central sub-space, it is possible to construct a C volume preserving diffeomorphism f whichis arbitrarily C -close to f and has negative central exponents on a set of positivevolume. Moreover, if C is a compact periodic central leaf, then f can be arrangedto coincide with f in a small neighborhood of the trajectory of C .We now consider the two particular examples. Example 1.
Consider the time-1 map f of the geodesic flow on a compactsurface of negative curvature. Clearly, f is partially hyperbolic and has a denseset of compact periodic central leaves. It follows from what was said above thatthere is a volume preserving perturbation f of f such that(1) f is of class C and is arbitrary close to f in the C -topology;(2) f is a partially hyperbolic diffeomorphism with one-dimensional centralsubspace;(3) there exists a central leaf C such that the restriction f ℓ |C is a minimaltransformation (here ℓ is the period of the leaf);(4) f has negative central exponents on a set of positive volume;(5) the unstable foliation for f is minimal and hence, satisfies Condition (D).We conclude that in this example the whole manifold is the homoclinic class ofevery hyperbolic periodic point of index two and that this class satisfies Conditions(H1) and (H2). Example 2.
Consider the map f = A × R of the 3-torus T = T × T where A is a linear Anosov automorphism of the 2-torus T and R is an irrational rotation ofthe circle T . It follows from what was said above that there is a volume preservingperturbation f of f such that the properties (1) – (5) in the previous examplehold, and hence the unique homoclinic class satisfies Conditions (H1) and (H2). Remark 2.1.
It was shown in [6] that the set of partially hyperbolic diffeomor-phisms with one dimensional central direction contains a C open and dense subsetof diffeomorphisms with minimal unstable foliation. However, in our examples weuse preservation of volume to ensure negative central Lyapunov exponents on aset of positive volume, so we cannot immediately apply the result in [6] to obtainan open set of systems for which Conditions (H1) and (H2) hold, compare withProblem 7.25 from [7]. Remark 2.2.
In both Examples 1 and 2 the map possesses a non-hyperbolic er-godic invariant measure (e.g. supported on the compact periodic leaf). We believethat in these examples presence of non-hyperbolic ergodic invariant measures ispersistent under small perturbations. Indeed, since the central subspace is one di-mensional, the central Lyapunov exponent with respect to a given ergodic measureis an integral of a continuous function (i.e., log of the expansion rate along thecentral subspace) over this measure, existence of periodic points of different indices
A. GORODETSKI AND YA. PESIN combined with (presumable) connectedness of the space of ergodic measures shouldimply existence of a non-hyperbolic invariant ergodic measure. See [3, 5, 14, 22] forthe related results and discussion.2.2.
Homoclinic classes that do not satisfy Conditions (H1) and (H2).
There is an example of an invariant set for a partially hyperbolic map with onedimensional central subspace which is a homoclinic class containing two non-homoclinically related hyperbolic periodic orbits of the same index, hence, notsatisfying Condition (H1), see [11, 12, 15]. Moreover, the space of hyperbolic er-godic measures supported on this homoclinic class is not connected due to thefact that the set of all central Lyapunov exponents is split into two disjoint closedintervals, see Remark 5.2 in [12].As we already mentioned in Introduction, Condition (H2) does not always holdeven for surface diffeomorphisms, see for example, [16, 19]. This condition ensuresthat the hyperbolic horseshoes and periodic orbits that we use to approximate agiven hyperbolic ergodic measure do belong to the initial homoclinic class. We donot know whether given a not necessarily isolated homoclinic class, every hyper-bolic ergodic invariant measure supported on this homoclinic class can always beapproximated in such a way. 3.
Proofs
The space M of all probability Borel measures on M equipped with the weak ∗ -topology is metrizable with the distance d M given by(1) d M ( µ, ν ) = ∞ X k =1 k (cid:12)(cid:12)(cid:12)(cid:12)Z ψ k dµ − Z ψ k dν (cid:12)(cid:12)(cid:12)(cid:12) , where { ψ k } k ∈ N is a dense subset in the unit ball in C ( M ). While the distancedefined in this way depends on the choice of the subset { ψ k } k ∈ N , the topology itgenerates does not. We will choose the functions ψ k to be smooth. Proof of Theorem 1.1.
By a hyperbolic periodic measure µ q we mean an atomicergodic measure equidistributed on a hyperbolic periodic orbit of q . Lemma 3.1.
Let q , q ∈ H ( p ) be hyperbolic periodic points with index s ( q ) = s ( q ) = s ( p ) . Then the hyperbolic periodic measures µ q and µ q can be connectedin M ep by a continuous path.Proof of Lemma 3.1. By Condition (H1), the points q and q are homoclinicallyrelated. By the Smale-Birkhoff theorem, there is a hyperbolic horseshoe Λ thatcontains both q and q . Lemma 3.1 now follows from the results by Sigmund, seethe proof of Theorem B in [32]. (cid:3) We wish to approximate a given hyperbolic measure by periodic measures. Thereare several results in this direction, see, for example, [2, Theorem 15.4.7]. However,we need some specific properties of such approximations that are stated in thefollowing lemma.
Lemma 3.2.
For any µ ∈ M ep and any ε > the following statements hold: By a hyperbolic horseshoe we mean a locally maximal hyperbolic set Λ which is totallydisconnected and such that f | Λ is topologically transitive.
ATH CONNECTEDNESS AND ENTROPY DENSITY 7 (1)
There exists a hyperbolic periodic point q ∈ H ( p ) with s ( q ) = s ( p ) such thatwe have d M ( µ q , µ ) < ε for the corresponding hyperbolic periodic measure µ q ; (2) There exists δ > such that for any hyperbolic periodic points q , q ∈ H ( p ) with s ( q ) = s ( q ) = s ( p ) if the corresponding hyperbolic periodic measures µ q and µ q lie in the δ -neighborhood of the measure µ , then there exists acontinuous path { ν t } t ∈ [0 , ⊂ M ep with ν = µ q , ν = µ q and such that d M ( ν t , µ ) < ε for all t ∈ [0 , .Proof of Lemma 3.2. Let R be the set of all Lyapunov-Perron regular points, andfor each ℓ ≥ R ℓ be the regular set (see [2] for definitions). There exists ℓ ∈ N such that µ ( R ℓ ) >
0. Fix ε >
0. By Birkhoff’s Ergodic Theorem, for a µ -genericpoint x ∈ R ℓ there exists N ∈ N such that for any n > N (2) d M n n − X k =0 δ f k ( x ) , µ ! < ε . Choose L ∈ N such that P ∞ k = L +1 12 k − < ε . Let { ψ k } be the dense collection ofsmooth functions { ψ k } in the definition (1) of the distance d M and let C = C ( ε )be the common Lipschitz constant of the functions { ψ , . . . , ψ L } . Let U ( H ( p )) bea neighborhood of H ( p ) such that H ( p ) = T n ∈ Z f n ( U ( H ( p ))); its existence is guar-anteed by (H2). Let us now choose δ > Cδ < ε and δ -neighborhoodof H ( p ) is in U . Since µ is a hyperbolic measure, by [2, Theorem 15.1.2], thereexists n > N and a hyperbolic periodic point y ∈ M of period n such thatdist M ( f k ( x ) , f k ( y )) < δ for all k = 0 , . . . , n − s ( y ) = s ( x ). Then the or-bit of y is in U , and hence belongs to H ( p ). Also, we have(3) d M n n − X k =0 δ f k ( x ) , n n − X k =0 δ f k ( y ) ! ≤ L X k =1 k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ψ k d n n − X i =0 δ f i ( x ) ! − Z ψ k d n n − X i =0 δ f i ( y ) !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + ∞ X k = L +1 k − ≤ Cδ + ε < ε . The first statement of the lemma now follows from (2) and (3).To prove the second statement let q , q ∈ H ( p ) be any hyperbolic periodicpoints such that s ( q ) = s ( q ) = s ( p ) and the corresponding hyperbolic periodicmeasures µ q and µ q lie in the δ -neighborhood of the measure µ . By Conditions(H1) and (H2), the points q and q are homoclinically related and hence, there isa hyperbolic horseshoe which contains both points. The desired result now followsfrom [32] (see the proof of Theorem B). (cid:3) We now compete the proof of Theorem 1.1. Let η and e η ∈ M ep be two hyperbolicergodic measures. By Statement 1 of Lemma 3.2, there are sequences of hyperbolicperiodic points q k and e q k in H ( p ) with s ( q k ) = s ( e q k ) = s ( p ) such that for thecorresponding sequences of hyperbolic periodic measures { µ q k } k ∈ N and { µ e q k } k ∈ N we have µ q k → η and µ e q k → e η . By Lemma 3.1, there is a path { ν t } t ∈ [ , ] in M ep that connects µ q and µ e q , that is, ν = µ q and ν = µ e q . By Statement 2 of A. GORODETSKI AND YA. PESIN
Lemma 3.2, for any k ∈ N there are paths { ν t } t ∈ [ k +1 , k ] and { ν t } t ∈ [ − k , − k +1 ]in M ep that connect measures µ q k , µ q k +1 and measures µ e q k , µ e q k +1 , respectively andthe length of each such path does not exceed ε k . Applying again Lemma 3.2, weconclude that the path { ν t } t ∈ [0 , given by the above choices and such that ν = η and ν = e η is continuous. The desired result now follows. (cid:3) Proof of Theorem 1.2.
Arguments similar to those used in the proof of Lemma 3.2(see also the proof of Theorem B in [32]) show that the following statement holds:
Lemma 3.3.
For any ε > there exists δ > such that for any two measures µ , µ ∈ M ep with d M ( µ , µ ) < δ there exists a continuous path in M ep connecting µ and µ of diameter smaller than ε . Now Theorem 1.2 can be obtained using Lemma 3.3 in the same way Theorem1.1 was obtained using Lemma 3.2. (cid:3)
Proof of Theorem 1.5.
Given a (not necessarily ergodic) measure µ ∈ M p , by theergodic decomposition, there exists a measure ν on the space M ep such that µ = Z τ dν ( τ ) and h µ = Z h τ dν ( τ ) . It follows that for any ε > τ , . . . , τ N ∈ M ep and positivecoefficients α , . . . , α N such that(4) d M p µ, N X k =1 α k τ k ! < ε and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h µ − N X k =1 α k h τ k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ε. Given a hyperbolic ergodic measure τ with h τ >
0, there exist a sequence of hyper-bolic horseshoes Λ n and a sequence of ergodic measures { ν n } n ∈ N supported on Λ n such that ν n → τ and h ν n → h τ as n → ∞ , see, for example, Corollary 15.6.2 in [2].By (H2), one can ensure in the construction of these horseshoes that Λ n ⊆ H ( p )and that ν n are the measures of maximal entropy and hence, Markov measures.Further, for every x ∈ Λ n the dimension of the stable manifold through x is equalto the index of p .In the case when h τ = 0 the measure τ can be approximated by a hyperbolicperiodic measure supported on an orbit of a hyperbolic periodic point q ∈ H ( p )(see Lemma 3.2) with s ( q ) = s ( p ). There exists a hyperbolic horseshoe Λ q ⊂ H ( p )that contains a periodic point q , and one can choose a Markov measure (which isnot a measure of maximal entropy in this case) supported on Λ q that is arbitraryclose to the atomic invariant measure supported on the orbit of q , and has arbirarysmall entropy (notice that the support of this Markov measure does not have to beclose to the orbit of q ).It follows from what was said above that to each hyperbolic ergodic measure τ k we can associate a hyperbolic horseshoe Λ k ⊆ H ( p ) and a Markov measure ν k supported on Λ k such that for every k = 1 , . . . , N we have(5) d M p ( τ k , ν k ) < εN and | h τ k − h ν k | < εN . Notice that all horseshoes Λ k have the same index s ( p ) and that they are homo-clinically related (this means that every periodic orbit in one of the horseshoes is ATH CONNECTEDNESS AND ENTROPY DENSITY 9 homoclinically related to any periodic orbit on the other horseshoe). This impliesthat there exists a hyperbolic horseshoe Λ ⊂ H ( p ) that contains all Λ k .The Markov measure ν k is constructed with respect to a Markov partition ofΛ k that we denote by ξ k . There exists a Markov partition ξ of Λ such that itsrestriction on each Λ k is a refinement of ξ k . The measure P Nk =1 α k ν k is a Markovmeasure on Λ with respect to the partition ξ . Notice that Markov measures aswell as their entropies depend continuously on their stochastic matrices. Therefore,given an arbitrarily (not necessarily ergodic) Markov measure, one can produce itssmall perturbation which is an ergodic Markov measure whose entropy is close tothe entropy of the unperturbed one. This gives the required approximation of themeasure P Nk =1 α k ν k , which by (4) and (5) is close to the initial measure µ andwhose entropy is close to h µ . (cid:3) References [1] A. Baraviera, C. Bonatti, Removing zero Lyapunov exponents,
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Manuscripta Math. (1977), 27–32. Department of Mathematics, University of California, Irvine, CA 92697, USA
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