Lyapunov spectrum for rational maps
LLYAPUNOV SPECTRUM FOR RATIONAL MAPS
KATRIN GELFERT, FELIKS PRZYTYCKI, AND MICHA(cid:32)L RAMS
Abstract.
We study the dimension spectrum of Lyapunov exponentsfor rational maps on the Riemann sphere.
Contents
1. Introduction 12. Tools for non-uniformly hyperbolic dynamical systems 52.1. Topological pressure 52.2. Building bridges between unstable islands 62.3. Hyperbolic subsystems and approximation of pressure 92.4. Conformal measures 112.5. Hyperbolic times and conical limit points 113. On the completeness of the spectrum 114. Upper bounds for the dimension 154.1. No critical points in J Introduction
Let f : C → C be a rational function of degree d ≥ J = J ( f ) be its Julia set. Our goal is to study the spectrumof Lyapunov exponents of f | J . Given x ∈ J we denote by χ ( x ) and χ ( x ) Mathematics Subject Classification.
Primary: 37D25, 37C45, 28D99 37F10 .
Key words and phrases.
Lyapunov exponents, multifractal spectra, iteration of rationalmap, Hausdorff dimension, nonuniformly hyperbolic systems.This research was supported by the EU FP6 Marie Curie programmes SPADE2 andCODY. The research of F. P and M. R. were supported by the Polish MNiSW GrantNN201 0222 33 ‘Chaos, fraktale i dynamika konforemna’. K. G. was partially supportedby the Deutsche Forschungsgemeinschaft and by the Humboldt foundation. K. G. andM. R. thank for the hospitality of MPI PKS Dresden and IM PAN Warsaw where part ofthis research was done. a r X i v : . [ m a t h . D S ] O c t KATRIN GELFERT, FELIKS PRZYTYCKI, AND MICHA(cid:32)L RAMS the lower and upper Lyapunov exponent at x , respectively, where χ ( x ) def = lim inf n →∞ n log | ( f n ) (cid:48) ( x ) | , χ ( x ) def = lim sup n →∞ n log | ( f n ) (cid:48) ( x ) | . If both values coincide then we call the common value the
Lyapunov exponent at x and denote it by χ ( x ). For given numbers 0 ≤ α ≤ β we consider alsothe following level sets L ( α, β ) def = (cid:8) x ∈ J : χ ( x ) = α, χ ( x ) = β (cid:9) . We denote by L ( α ) def = L ( α, α ) the set of Lyapunov regular points withexponent α . If α < β then L ( α, β ) is contained in the set of so-called irregular points L irr def = (cid:8) x ∈ J : χ ( x ) < χ ( x ) (cid:9) . Recall that it follows from the Birkhoff ergodic theorem that µ ( L irr ) = 0 forany f -invariant probability measure µ .While the first results on the multifractal formalism go already back toBesicovitch [4], its systematic study has been initiated by work of Collet,Lebowitz and Porzio [5]. The case of spectra of Lyapunov exponents forconformal uniformly expanding repellers has been covered for the first timein [2] building also on work by Weiss [26] (see [13] for more details andreferences). To our best knowledge, the first results on irregular parts ofa spectrum were obtained in [4]. Its first complete description (for digitexpansions) was given by Barreira, Saussol, and Schmeling [3].In this work we will formulate our results on the spectrum of Lyapunovexponents in terms of the topological pressure P . For any α > F ( α ) def = 1 α inf d ∈ R (cid:0) P f | J ( − d log | f (cid:48) | ) + dα (cid:1) . and F (0) def = lim α → F ( α ) . Let [ α − , α + ] be the interval on which F (cid:54) = −∞ (a more formal and equivalentdefinition is given in (4) below).Before stating our main results, we recall what is already known in thecase that J is a uniformly expanding repeller with respect to f , that is, J is a compact f -forward invariant (i.e., f ( J ) = J ) isolated set such that f | J is uniformly expanding. Recall that f is said to be uniformly expanding or uniformly hyperbolic on a set Λ if there exist c > λ > n ≥ x ∈ Λ we have | ( f n ) (cid:48) | ≥ cλ n . Recall that a set Λ issaid to be isolated if there exists an open neighborhood U ⊂ C of Λ suchthat f n ( x ) ∈ U for every n ≥ x ∈ Λ. In our setting the Julia set J is a uniformly expanding repeller if it does not contain any critical pointnor parabolic point. Here a point x is said to be critical if f (cid:48) ( x ) = 0 and to N THE LYAPUNOV SPECTRUM FOR RATIONAL MAPS 3 be parabolic if x is periodic and its multiplier ( f per( x ) ) (cid:48) ( x ) is a root of 1. If J is a uniformly expanding repeller then for every α ∈ [ α − , α + ] we havedim H L ( α ) = F ( α )(see [2, 26]) and L ( α ) = ∅ if and only if α / ∈ [ α − , α + ] [23]. This gives the fulldescription of the regular part of the Lyapunov spectrum. Moreover, in thissetting of a uniformly expanding repeller J , the interval [ α − , α + ] coincideswith the closure of the range of the function α ( d ) = − dds P f | J ( − s log | f (cid:48) | ) | s = d and the spectrum can be written as F ( α ( d )) = 1 α ( d ) (cid:0) P f | J ( − d log | f (cid:48) | ) + dα ( d ) (cid:1) = h µ d ( f ) α ( d ) , where α ( d ) is the unique number satisfying − dds P f | J ( − s log | f (cid:48) | ) | s = d = (cid:90) log | f (cid:48) | dµ d = α ( d )and where µ d is the unique equilibrium state corresponding to the potential − d log | f (cid:48) | . If log | f (cid:48) | is not cohomologous to a constant then we have α − <α + , and α (cid:55)→ αF ( α ) and d (cid:55)→ P f | J ( − d log | f (cid:48) | ) are real analytic strictlyconvex functions that form a Legendre pair.We now state our first main result. Theorem 1.
Let f be a rational function of degree ≥ with no criticalpoints in its Julia set J . For any α − ≤ α ≤ β ≤ α + , β > , we have dim H L ( α, β ) = min { F ( α ) , F ( β ) } . In particular, for any α ∈ [ α − , α + ] \ { } we have dim H L ( α ) = F ( α ) . If there exists a parabolic point in J (and hence F (0) > −∞ ) then dim H L (0) = dim H J = F (0) . Moreover, (cid:8) x ∈ J : χ ( x ) < α − (cid:9) = (cid:8) x ∈ J : χ ( x ) > α + (cid:9) = ∅ . We denote by Crit the set of all critical points of f . Following Makarovand Smirnov [10, Section 1.3], we will say that f is exceptional if there existsa finite, nonempty set Σ f ⊂ C such that f − (Σ f ) \ Crit = Σ f . This set need not be unique. We will further denote by Σ the largest of suchsets (notice that it has no more than 4 points).
Theorem 2.
Let f be a rational function of degree ≥ . Assume that f is non-exceptional or that f is exceptional but Σ ∩ J = ∅ . For any < α ≤ β ≤ α + we have min { F ( α ) , F ( β ) } ≤ dim H L ( α, β ) ≤ max α ≤ q ≤ β F ( q ) . KATRIN GELFERT, FELIKS PRZYTYCKI, AND MICHA(cid:32)L RAMS
In particular, for any α ∈ [ α − , α + ] \ { } we have dim H L ( α ) = F ( α ) and dim H L (0) ≥ F (0) . Moreover, (cid:8) x ∈ J : − ∞ < χ ( x ) < α − (cid:9) = (cid:8) x ∈ J : χ ( x ) > α + (cid:9) = ∅ and dim H (cid:8) x ∈ J : 0 < χ ( x ) < α − (cid:9) = 0 . If f is exceptional and Σ ∩ J (cid:54) = ∅ (this happens, for example, for Cheby-shev polynomials and some Latt`es maps) then the situation can be much dif-ferent from the above-mentioned cases. For example, the map f ( x ) = x − −∞ , two pointswith Lyapunov exponent 2 log 2, a set of dimension 1 of points with Lya-punov exponent log 2, and no other Lyapunov regular points. Hence, forthis map the Lyapunov spectrum is not complete in the interval [ α − , α + ] =[log 2 , ∩ J = ∅ . We do notknow how big the set L ( −∞ ) is except in the case when f has only onecritical point in J (in which case L ( −∞ ) consists only of the backwardorbit of this critical point). Moreover, we do not know whether the set { x ∈ J : χ ( x ) < α − } contains any points other than the backward orbitsof critical points contained in J and we only have some estimation for theHausdorff dimension of the set L ( α, β ) even for values α , β ∈ [ α − , α + ].The paper is organized as follows. In Section 2 we introduce the toolswe are going to use in this paper. In particular, we construct a family ofuniformly expanding Cantor repellers with pressures pointwise convergingto the pressure on J (Proposition 1). Section 3 discusses general propertiesof the spectrum of exponents. In Section 4 we obtain upper bounds for theHausdorff dimension. Here we use conformal measures to deal with coni-cal points (Proposition 2) and we prove that the set of non-conical pointswith positive upper Lyapunov exponent is very small using the pullbackconstruction (Proposition 3). In Section 5 we derive lower bounds for thedimension. To do so, we first consider the interior of the spectrum and wewill use the sequence of Cantor repellers from Section 2 to obtain for any α ∈ ( α − , α + ) a big uniformly expanding subset of points with Lyapunov ex-ponent α from which we derive our estimates. We finally study the boundaryof the spectrum and the irregular part of the spectrum using a construction,that generalizes the w-measure construction from [8]. N THE LYAPUNOV SPECTRUM FOR RATIONAL MAPS 5 Tools for non-uniformly hyperbolic dynamical systems
Topological pressure.
Given a compact f -invariant set Λ ⊂ J , wedenote by M (Λ) the family of f -invariant Borel probability measures sup-ported on Λ. We denote by M E (Λ) the subset of ergodic measures. Given µ ∈ M E (Λ), we denote by χ ( µ ) def = (cid:90) Λ log | f (cid:48) | dµ the Lyapunov exponent of µ . Notice that we have χ ( µ ) ≥ µ ∈ M ( J ) [17].Given d ∈ R , we define the function ϕ d : J \ Crit → R by(1) ϕ d ( x ) def = − d log | f (cid:48) ( x ) | . Given a compact f -invariant uniformly expanding set Λ ⊂ J , the topologicalpressure of ϕ d (with respect to f | Λ ) is defined by(2) P f | Λ ( ϕ d ) def = max µ ∈ M (Λ) (cid:18) h µ ( f ) + (cid:90) Λ ϕ d dµ (cid:19) , where h µ ( f ) denotes the entropy of f with respect to µ . We simply write M = M ( J ) and P ( ϕ d ) = P f | J ( ϕ d ) if we consider the full Julia set J andif there is no confusion about the system. A measure µ ∈ M is called equilibrium state for the potential ϕ d (with respect to f | J ) if P ( ϕ d ) = h µ ( f ) + (cid:90) J ϕ d dµ. For every d ∈ R we have the following equivalent characterizations of thepressure function (see [21], where further equivalences are shown). We have P ( ϕ d ) = sup µ ∈ M +E (cid:18) h µ ( f ) + (cid:90) J ϕ d dµ (cid:19) = sup Λ P f | Λ ( ϕ d ) . (3)Here in the first equality the supremum is taken over the set M +E of all ergodic f -invariant Borel probability measures on J that have a positive Lyapunovexponent and are supported on some f -invariant uniformly expanding subsetof J . In the second equality the supremum is taken over all uniformlyexpanding repellers Λ ⊂ J . In fact, in the second equality it suffices totake the supremum over all uniformly expanding Cantor repellers, that is,uniformly expanding repellers that are limit sets of finite graph directedsystems satisfying the strong separation condition with respect to f , seeSection 2.2. KATRIN GELFERT, FELIKS PRZYTYCKI, AND MICHA(cid:32)L RAMS
Let us introduce some further notation. Let α − def = lim d →∞ − d P ( ϕ d ) = inf µ ∈ M χ ( µ ) ,α + def = lim d →−∞ − d P ( ϕ d ) = sup µ ∈ M χ ( µ ) , (4)where the given characterizations follow easily from the variational principle.Recall that, given α >
0, we define(5) F ( α ) def = 1 α inf d ∈ R ( P ( ϕ d ) + αd )and(6) F (0) def = lim α → F ( α ) . Note that F (0) = d def = inf { d : P ( ϕ d ) = 0 } . Building bridges between unstable islands.
We describe a con-struction of connecting two given hyperbolic subsets of the Julia set by“building bridges” between the sets. We call a point x ⊂ J non-immediately post-critical if there exists somepreimage branch x = x = f ( x ), x = f ( x ), . . . that is dense in J anddisjoint from Crit. If f is non-exceptional or if it is exceptional but Σ ∩ J = ∅ then for every hyperbolic set all except possibly finitely many points (inparticular, all periodic points) are non-immediately post-critical.We will now consider a set Λ that is an f -uniformly expanding Cantorrepeller (ECR for short), that is, a uniformly expanding repeller and a limitset of a finite graph directed system (GDS) satisfying the strong separationcondition (SSC) with respect to f . Recall that by a GDS satisfying the SSCwith respect to f we mean a family of domains and maps satisfying thefollowing conditions (compare [12, pp. 3, 58]):(i) There exists a finite family U = { U k : k = 1 , ..., K } of open connected(not necessarily simply connected) domains in the Riemann spherewith pairwise disjoint closures.(ii) There exists a family G = { g k(cid:96) : k, (cid:96) ∈ { , ..., K }} of branches of f − mapping U (cid:96) into U k with bounded distortion (not all pairs k, (cid:96) mustappear here).Note that a general definition of GDS allows many maps g from each U (cid:96) to each U k . Here however there can be at most one, since weassume that f -critical points are far away from Λ and that the maps g are branches of f − . This is a precise realization of an idea of Prado [15].
N THE LYAPUNOV SPECTRUM FOR RATIONAL MAPS 7 (iii) We have Λ = ∞ (cid:92) n =1 (cid:91) k ,...,k n g k k ◦ g k k ◦ ... ◦ g k n − k n . We assume that we have f (Λ) = Λ and hence that for each k thereexists (cid:96) and for each (cid:96) there exists k such that g k(cid:96) ∈ G .We can view k = 1, ... , K as vertices and g k(cid:96) as edges from (cid:96) to k of adirected graph Γ = Γ( U , G ).This definition easily implies that f is uniformly expanding on the limitset Λ of such a GDS, and that Λ is a repeller for f . Clearly Λ is a Cantor set.In fact a sort of converse is true (though we shall not use this fact in thispaper, but it clears up the definitions). Namely we observe the followingfact. Lemma 1. If Λ ⊂ J is an f -invariant compact uniformly expanding setthat is a Cantor set, then Λ is contained in the limit set of a GDS satisfyingthe SSC (this limit set can be chosen to be contained in an arbitrarily smallneighborhood of Λ ). Hence, Λ is contained in an f -ECR set.Proof. We can multiply the standard sphere Riemann metric by a positivesmooth function such that with respect to this new metric ρ Λ we have | f (cid:48) | ≥ λ > r > B (Λ , r ) = { z ∈ C : ρ Λ ( z, Λ) < r } consistsof a finite number of connected open domains U k with pairwise disjointclosures. We account for our GDS the branches of g = f − on the sets U k such that each g ( U k ) intersects Λ. Then g maps each U k into some U (cid:96) because it is a contraction (by the factor λ − ). Hence the family of maps g | U k satisfies the assumptions of a GDS with the SSC. (cid:3) In the proof of the following lemma we will “build bridges” between twoECR’s.
Lemma 2.
For any two disjoint f -ECR sets Λ , Λ ⊂ J that both containnon-immediately postcritical points there exists an f -ECR set Λ ⊂ J con-taining the set Λ ∪ Λ . If f is topologically transitive on each Λ i , i = 1 , ,then f is topologically transitive on Λ .Proof. Let Λ , Λ ⊂ J be two sets satisfying the assumptions of the lemmaand let p ∈ Λ and p ∈ Λ be two non-immediately postcritical points.Consider a family U i = { U i,k } K i k =1 of open connected domains and a family G i = { g i,k(cid:96) } of branches of f − mapping U i,(cid:96) to U i,k that define the DGS’ssatisfying the SSC that have Λ i as their limit sets, for i = 1, 2, respectively.Let D i def = K i (cid:91) k =1 U i,k . KATRIN GELFERT, FELIKS PRZYTYCKI, AND MICHA(cid:32)L RAMS
We can assume that each D i is an arbitrarily small neighborhood of Λ i , byreplacing U i by G m ( U i ), where by G m we denote the family of all composi-tions G m = (cid:8) g i,k k ◦ g i,k k ◦ ... ◦ g i,k m − k m : g i,k n k n +1 ∈ G i , n = 1 , . . . , m − (cid:9) . For each i = 1, 2 let us choose a backward trajectory y i,t of the point p i (the“bridge”) such that y i, = p i , f ( y i,t ) = y i,t − for t = 1 , ..., t i ,y i,t / ∈ D ∪ D for all t = 1, . . . , t i − y ,t ∈ D , y ,t ∈ D . Let usdenote by h i,t the branch of f − t that maps p i to y i,t , that is, let h i,t def = f − ty i,t . Let V i be an open disc centered at p i that is contained in D i and satisfies h i,t ( V i ) ∩ ( D ∪ D ) = ∅ for all t = 1, . . . , t i − V i small enough) and h ,t ( V ) ⊂ D and h ,t ( V ) ⊂ D . Let us consider an integer N ≥ f − N ( D i )containing p i is contained in V i , that is, that f − Np i ( D i ) ⊂ V i for i = 1, 2. Now let us replace U i by (cid:98) U i def = G N ( U i ), let us replace each map g i,k(cid:96) by the family of its restrictions to (cid:98) U ∈ (cid:98) U i contained in U (cid:96) and and letus denote by (cid:98) G i the union of those families. This defines a GDS with graphΓ i = Γ i ( (cid:98) U i , (cid:98) G i ), for i = 1, 2. Now we restrict each bridge h i,t to the element (cid:98) V i of (cid:98) U i that contains p i . As the next step we consider (cid:98) V i,t def = h i,t ( (cid:98) V i ) for t = 1 , ..., t i + N − , where for t > t i we choose an arbitrary prolongation of the bridge y i,t by maps g j,k t k t +1 . Finally, we consecutively thicken slightly (cid:98) V i,t along thebridges such that f ( (cid:98) V i,t ) ⊃ (cid:98) V i,t − . For each t let us denote by g i,t the branchof f − from (cid:98) V i,t to (cid:98) V i,t +1 for t = 0, ... , t i + N −
1. Let us denote by H i thefamily of all these branches. By construction the family of domains U def = (cid:98) U ∪ (cid:98) U ∪ (cid:91) i =1 , (cid:91) t =1 ,...,t i + N − (cid:98) V i,t and the family of maps G def = (cid:98) G ∪ (cid:98) G ∪ H ∪ H form our desired GDS witha graph Γ = Γ( U , G ) satisfying the SSC and hence defines an f -ECR setΛ ⊂ J that containes Λ ∪ Λ .Finally notice that this system has topologically transitive limit set Λsince its transition graph Γ is transitive. This follows from the assumptionthat due to topological transitivity of f | Λ i the graphs Γ i are transitive andfrom the construction of the bridges. (cid:3) N THE LYAPUNOV SPECTRUM FOR RATIONAL MAPS 9
Hyperbolic subsystems and approximation of pressure.
Ourapproach is to “exhaust” the Julia set J by some family of subsets Λ m ⊂ J and to show that the corresponding pressure functions converge towards thepressure of f | J . In particular, in order to be able to conclude convergence ofassociated spectral quantities, it is crucial that each such Λ m is an invariantuniformly expanding and topologically transitive set.We start by stating a classical result from Pesin-Katok theory. In followsfor example from [22, Theorems 10.6.1 and 11.2.3]. Recall that an iteratedfunction system (IFS) is a GDS that is given by a complete graph. Lemma 3.
For every ergodic f -invariant measure µ that is supported on J and has a positive Lyapunov exponent, for every continuous function φ : J → R and for every ε > , there exist an integer n > and an f n -ECR set Λ ⊂ J that is topologically transitive and a limit set of an IFS, such that dim H Λ ≥ dim H µ − ε (7) P f n | Λ ( S n φ ) ≥ h µ ( f n ) + n (cid:90) φ dµ − nε, where we use the notation S n φ ( x ) def = φ ( x ) + φ ( f ( x )) + . . . + φ ( f n − ( x )) , andin particular (8) P f | (cid:83) n − k =0 f k (Λ) ( φ ) ≥ h µ ( f ) + (cid:90) φ dµ − ε. Our aim is to apply Lemma 3 to potentials φ = ϕ d and to use the resultingECR sets to construct a sequence of f a m -ECR sets Λ m on which the pressurefunction a m P f am | Λ m ( S a m ϕ d ) converges pointwise to P ( ϕ d ). If the ECR setsgenerated by Lemma 3 are pairwise disjoint, we can simply build bridgesbetween such sets applying Lemma 2. In the general situation we start withthe following lemma. Lemma 4.
Let Λ be a topologically transitive f n -ECR set. Let φ i : Λ → R be a finite number of continuous functions. Then for any open disc D intersecting Λ and for any ε > one can find a set Λ (cid:48) ⊂ D ∩ Λ and a naturalnumber (cid:96) > such that Λ (cid:48) is an f (cid:96) -ECR set and for every φ i satisfies (cid:96) P f (cid:96) | Λ (cid:48) ( S (cid:96) φ i ) ≥ n P f n | Λ ( S n φ i ) − ε. Proof.
Consider in Λ a clopen set C contained in D . Let x ∈ C . Let uschoose N large enough such that in the case that (cid:96) ≥ N and f − (cid:96)x ( C ) ∩ C (cid:54) = ∅ the pullback satisfies f − (cid:96)x ( C ) ⊂ C . Note that it is enough to take N > log diam Λ ρ Λ ( C, Λ \ C ) log λ , where λ is the expanding constant on Λ in the metric ρ Λ defined as in theproof of Lemma 1. Note that the topological transitivity of f n | Λ implies thatevery pullback of C can be continued by a bounded number of consecutivepullbacks until it hits C , say this number is bounded by a constant N (cid:48) . This way we obtain an IFS for f (cid:96) , where N ≤ (cid:96) ≤ N + N (cid:48) , with its limit set Λ (cid:48) contained in C .Recall the equivalent definition of tree pressure established in [21, The-orems A, A.4]. Due to topological transitivity in the definition of pressurewe can consider separated sets that are contained in the set of preimages f − N ( x ). Therefore, for any φ i the pressures with respect to f | Λ (cid:48) and withrespect to f | Λ differ by at most O ( N (cid:48) N ) from each other. As N can be chosenarbitrarily big, this proves the lemma. (cid:3) The following approximation results are fundamental for our approach.
Proposition 1.
Assume that f is non-exceptional or that f is exceptionalbut Σ ∩ J = ∅ . Then there exists a sequence { a m } m of positive integersand a sequence Λ m ⊂ J of f a m -invariant uniformly expanding topologicallytransitive sets such that for every d ∈ R , we have (9) P ( ϕ d ) = lim m →∞ a m P f am | Λ m ( S a m ϕ d ) = sup m ≥ a m P f am | Λ m ( S a m ϕ d ) . For every α ∈ ( α − , α + ) we have (10) F ( α ) = lim m →∞ F m ( α ) = sup m ≥ F m ( α ) and (11) lim m →∞ α − m = inf m ≥ α − m = α − , lim m →∞ α + m = sup m ≥ α + m = α + , where F m and α ± m are defined as in (5) and (4) but with a m P f am | Λ m ( S a m φ ) instead of P f | J ( φ ) .Proof. To prove (9) it is enough to construct an f a m -ECR set Λ m ⊂ J suchthat(12) 1 a m P f am | Λ m ( S a m ϕ d ) ≥ P ( ϕ d ) − m for all d ∈ [ − m, m ]. Recall that we have (3). As d (cid:55)→ P ( ϕ d ) is uniformly Lip-schitz continuous, we only need to check (12) for a finite number of potentials φ i = ϕ d i . Given m , we apply Lemma 3 to potentials φ i , obtaining a familyof f n i -ECR sets Λ m,i on which the pressure n i P f ni | Λ m,i ( S n i φ i ) ≥ P ( φ i ) − m . We then apply Lemma 4 to construct a family of pairwise disjoint f (cid:96) i -ECR sets Λ (cid:48) m,i that satisfy (cid:96) i P f (cid:96)i | Λ (cid:48) m,i ( S (cid:96) i φ i ) ≥ P ( φ i ) − m . Those setsΛ (cid:48) m,i are also disjoint f a m -ECR sets for a m = (cid:81) i (cid:96) i and by our assumptioncontain non-immediately post-critical points. Hence we can consecutivelyapply Lemma 2 to them. We obtain an f a m -ECR set satisfying (12) for all φ i . This proves (9).As d (cid:55)→ P ( ϕ d ) and α (cid:55)→ αF ( α ) form a Legendre pair, (10) and (11) followfrom (9) by a result of Wijsman [27]. (cid:3) N THE LYAPUNOV SPECTRUM FOR RATIONAL MAPS 11
Remark 1.
It is enough for us to work with hyperbolic sets for some it-erations of f instead of hyperbolic sets for f (in other words, to use (7)instead of (8)). However, notice that we could instead also extend the set (cid:83) a m − i =0 f i (Λ m ) to an f -ECR set using Lemma 1.2.4. Conformal measures.
The dynamical properties of any measure ν with respect to f | J are captured through its Jacobian. The Jacobian of ν with respect to f | J is the (essentially) unique function Jac ν f determinedthrough(13) ν ( f ( A )) = (cid:90) A Jac ν f dν for every Borel subset A of J such that f | A is injective. In particular, itsexistence yields the absolute continuity ν ◦ T ≺ ν .A probability measure ν that satisfiesJac ν f = e P ( ϕ d ) − ϕ d is called e P ( ϕ d ) − ϕ d -conformal measure . If d ≥ e P ( ϕ d ) − ϕ d -conformal measure ν d that is positive on each open set intersecting J , see [21]. When d < f is not anexceptional map or if f is exceptional but Σ ∩ J = ∅ (see [21, AppendixA.2]).2.5. Hyperbolic times and conical limit points.
When f | J is not uni-formly expanding, we can still observe a slightly weaker form of non-uniformhyperbolicity. We recall two concepts that have been introduced.A number n ∈ N is called a hyperbolic time for a point x with exponent σ if | ( f k ) (cid:48) ( f n − k ( x )) | ≥ e kσ for every 1 ≤ k ≤ n. It is an immediate consequence of the Pliss lemma (see, for example, [14])that for a given point x ∈ J , for any σ < χ ( x ) there exist infinitely manyhyperbolic times for x with exponent σ .We denote by Dist g | Z def = sup x,y ∈ Z | g (cid:48) ( x ) || g (cid:48) ( y ) | the maximal distortion of a map g on a set Z . After [6], we will call a point x ∈ J conical if there exist a number r >
0, a sequence of numbers n i (cid:37) ∞ and a sequence { U i } i of neighborhoods of x such that f n i ( U i ) ⊃ B ( f n i ( x ) , r )and that Dist f n i | U i is bounded uniformly in i .3. On the completeness of the spectrum
In the following two lemmas we will investigate which numbers can occurat all as upper/lower Lyapunov exponents.
Lemma 5.
We have (cid:8) x ∈ J : χ ( x ) > α + (cid:9) = ∅ . If J does not contain any critical point of f then we have (cid:8) x ∈ J : χ ( x ) < α − (cid:9) = ∅ . Proof.
Consider an arbitrary x ∈ J and a sequence n i (cid:37) ∞ such thatlim i →∞ n i n i − (cid:88) j =0 log | f (cid:48) ( f j ( x )) | = χ ( x )and µ i def = 1 n i n i − (cid:88) j =0 δ f j ( x ) → µ in the weak ∗ topology. The limit measure µ is f -invariant, [25, Theorem6.9]. Define g N def = max { log | f (cid:48) | , − N } . Notice that { g N } N is a monotonically decreasing sequence of continuousfunctions that converge pointwise to log | f (cid:48) | . Hence we obtain χ ( x ) ≤ lim N →∞ (cid:90) g N dµ = (cid:90) log | f (cid:48) | dµ = χ ( µ ) ≤ α + , where the equality follows from the Lebesgue monotone convergence theo-rem. This proves the first statement.The second statement follows simply from the fact that log | f (cid:48) | is contin-uous on J if f has no critical points in J . (cid:3) Remark 2.
We remark that the same method of proof gives a slightlystronger result than the fact that χ ( x ) ≤ α + for every x . Namely we havelim n →∞ sup z ∈ C n log | ( f n ) (cid:48) ( z ) | ≤ α + , see [19, Proposition 2.3. item 2].Recall that on a set of total probability we have χ ( x ) ≥ x (comparethe proof of [20, Proposition 4.1]). Lemma 6. If x ∈ J has a finite Lyapunov exponent χ ( x ) then χ ( x ) ≥ α − ,that is, we have (cid:8) x ∈ J : − ∞ < χ ( x ) < α − (cid:9) = ∅ . If there are no critical points in J then L ( −∞ ) is empty. If there is onlyone critical point in J then L ( −∞ ) consists only of this critical point andits preimages. N THE LYAPUNOV SPECTRUM FOR RATIONAL MAPS 13
Proof.
Let x ∈ J be a Lyapunov regular point with exponent χ ( x ) and as-sume that χ ( x ) < α − . It is enough for us to prove that there exists a periodicpoint with Lyapunov exponent arbitrarily close to χ ( x ), the contradictionwill then follow from the definition of α − .Note first that if χ ( x ) exists and is finite then n log | ( f n ) (cid:48) ( x ) | must be aCauchy sequence and hence satisfies(14) lim inf n →∞ n log ρ ( f n ( x ) , Crit) = 0 . By [17, Corollary to Lemma 6], we then can conclude that χ ( x ) ≥
0. Choosenow a small number δ >
0. Let n be a hyperbolic time for x with exponent − δ/ x has infinitely many hyperbolic times with that exponent,and hence that n can be chosen arbitrarily big). Because of (14), there existsan integer n > k ≥ n we have(15) ρ ( f k ( x ) , Crit) > exp( − kδ )and we can assume that n > n .We start with a construction that is standard in Pesin theory. For each k = 0, . . . , n − n − B k def = B (cid:16) f n − k ( x ) , e ( − ( d +1) n + k ) δ (cid:17) , where d is the greatest degree of a critical point of f . This way we definea sequence of balls that are centered at points of the backward trajectoryof f n ( x ) and that have diameters shrinking slower than the derivative of f − k along this branch and at the same time have diameters much smallerthan their distance from any critical point. As we have f n − k ( x ) ∈ B k , (15)implies that for n big enough for any k < n − n the set f − f n − k − ( x ) ( B k ) doesnot contain any critical point and that(16) log Dist f − f n − k − ( x ) | B k ≤ K diam B k ρ ( f n − k ( x ) , Crit) , where K is some constant that depends only on f . Claim: If n is sufficiently big then for any k = 0, . . . , n the map f k isunivalent and has bounded distortion ≤ exp( δ/
2) on the set f − kf n − k ( x ) ( B )and satisfies f k ( B k ) ⊃ B .To prove the above claim, note that the ball B n − k shrinks as n increases.Hence, it is enough to prove the statement for k < n − n . The statementfor the initial finitely many steps k = n − n , . . . , n is then automaticallyprovided n is big enough. Let us assume that n is sufficiently big such thatalso K ( n − n ) e − dnδ < δ . By construction of the family { B k } k and by (15), for each (cid:96) ≥ (cid:96) − (cid:88) k =0 diam B k ρ ( f n − k ( x ) , Crit) < (cid:96) − (cid:88) k =0 e ( − ( d +1) n + k ) δ e ( − n + k ) δ = (cid:96)e − dnδ . Hence, if for (cid:96) ≤ n − n we have f k ( B k ) ⊃ B for every k = 1, . . . , (cid:96) − f − (cid:96)f n − (cid:96) ( x ) | B ≤ K (cid:96)e − dnδ < δ . On the other hand, recall that n is a hyperbolic time for x with exponent − δ/
2. Hence, if log Dist f − (cid:96)f n − (cid:96) ( x ) | B < δ f (cid:96) ( B (cid:96) ) ⊃ B . The above claim now follows by induction over (cid:96) .Let us consider the set E def = (cid:92) k ≥ (cid:91) (cid:96)>k E (cid:96) , where E (cid:96) def = (cid:96) (cid:91) j =1 B (cid:0) f j (Crit) , e − d(cid:96)δ (cid:1) . Notice that B \ E n is nonempty whenever the hyperbolic time n is bigenough. For such n , let y ∈ B \ E n . From our distortion estimations for f − nx | B in the Claim we obtain(17) log | ( f − nx ) (cid:48) ( f n ( x )) | (cid:12)(cid:12) ( f − nx ) (cid:48) ( y ) (cid:12)(cid:12) < δ . In the remaining proof we will follow closely techniques in [20, Section3]. Let us fix some arbitrary f -invariant uniformly expanding set Λ ⊂ J that has positive Hausdorff dimension (the existence of such a set followsfor example from Lemma 3). As Crit is a finite set, we have dim Λ E = 0and we can find a point z ∈ Λ \ E . In particular z ∈ Λ \ E n for n large. Notethat in particular for every n large enough we have(18) ρ ( z, f n (Crit)) > e − dnδ . and hence on the disk B ( z, e − dnδ ) any pull-back f − n is univalent.By [18, Lemma 3.1], there exists a number K > f and a sequence of disks { D i } i =1 ,...,K such that (cid:83) Ki =1 D i is connected, y is thecenter of disk D , z is the center of disk D K , ρ (cid:16) D i , n (cid:91) j =1 f j (Crit) (cid:17) ≥ diam D i , and the number of disks is bounded by K ≤ K ( nδ ) / . By the Koebedistortion lemma, for each branch of f − n and for every D i we haveDist f − n | D i < K , N THE LYAPUNOV SPECTRUM FOR RATIONAL MAPS 15 where K > (cid:12)(cid:12) log | ( f − nw ) (cid:48) ( z ) | − log | ( f − nx ) (cid:48) ( y ) | (cid:12)(cid:12) ≤ K log K for some w ∈ f − n ( z ). Together with (17), this implies that1 n (cid:12)(cid:12)(cid:12) log | ( f n ) (cid:48) ( x ) | − log | ( f n ) (cid:48) ( w ) | (cid:12)(cid:12)(cid:12) ≤ n (cid:18) δ K log K (cid:19) ≤ n (cid:18) δ K ( nδ ) / log K (cid:19) < δ , whenever n has been chosen large enough.By hyperbolicity of f | Λ , there exist positive constants ∆ and K such thatfor all (cid:96) ≥ f − (cid:96) is univalent and has bounded distortion ≤ K on B ( f (cid:96) ( z ) , ∆). Recall that the Julia set J has the property that there exists m = m (∆) > f m ( B ( f (cid:96) ( z ) , ∆) ∩ J ) is equal to J .Let (cid:96) be the smallest positive integer such that | ( f (cid:96) ) (cid:48) ( z ) | ≥ K ∆ e d ( n + m ) δ .Hence f − (cid:96)z ( B ( f (cid:96) ( z ) , ∆)) ⊂ B ( z, e − d ( n + m ) δ ) and it is easy to show that (cid:96) ≤ K + K ( n + m ) δ for some constants K , K . So in particular, oneof the preimages f − ( n + m ) ( z ) is in B ( f (cid:96) ( z ) , ∆) and the corresponding pull-back W def = f − ( n + m ) ( B ( z, e − d ( n + m ) δ )) satisfies W ⊂ B ( f (cid:96) ( z ) , ∆). It followsfrom (18) that f − ( n + m + (cid:96) ) : W → B ( z, e − n + m ) δ ) is univalent and hence thatthere is a repelling fixed point p = f n + m + (cid:96) ( p ) in B ( z, e − ( n + m + (cid:96) ).Using the above, the Lyapunov exponent of p can be estimated by χ ( p ) = 1 n + m + (cid:96) log | ( f n + m + (cid:96) ) (cid:48) ( p ) |≤ n log (cid:12)(cid:12) ( f n ) (cid:48) ( w ) (cid:12)(cid:12)
11 + ( m + (cid:96) ) /n + m + (cid:96)n + m + (cid:96) sup log | f (cid:48) |≤ (cid:18) χ ( x ) + δ (cid:19)
11 + ( m + (cid:96) ) /n + m + (cid:96)n + m + (cid:96) sup log | f (cid:48) | . Now recall that the hyperbolic time n can be chosen arbitrarily large andthat δ was chosen arbitrarily small. This proves the first claim of the lemma.If there are no critical points in J then log | f (cid:48) | (and hence χ ) is boundedfrom below. The remaining claim of the lemma follows from [7, Lemmas 2.1and 2.3]. (cid:3) Upper bounds for the dimension
In this section we will derive upper bounds for the Hausdorff dimensionof the level sets. We first start with a particular case.4.1.
No critical points in J . We study the particular case that there is nocritical point in the Julia set J (though, parabolic points in J are allowed).We start by taking a more general point of view and investigate thosepoints that have some least degree of hyperbolicity. In terms of Lyapunovexponents, this concerns points x with χ ( x ) >
0. We begin our analysis bypresenting a simple lemma, that will be useful shortly after.
Lemma 7.
Let { a n } n be a sequence of real numbers such that { a n +1 − a n } n converges to zero but { a n } n does not have a limit. Then for any naturalnumber r and for any number q ∈ [lim inf n →∞ a n , lim sup n →∞ a n ] there ex-ists a subsequence n k (cid:37) ∞ such that lim k →∞ a n k = q and for every k we have a n k < a n k + r . Proof.
We will restrict our hypothesis to the case that r = 1. The generalcase then follows from considering the subsequence { a rn } n .First note that every number q ∈ [lim inf a n , lim sup a n ] is the limit ofsome subsequence of { a n } n . Hence, if q (cid:54) = lim inf a n then for every ε > m = m ( ε ) such that a m < q − ε < a m +1 . Similarly, if q (cid:54) = lim sup a n then for every ε > m = m ( ε ) such that a m < q + ε < a m +1 . Since we assume that the sequence { a n } n does not have a limit, q must satisfyone of the abovementioned properties. Hence, if we choose a decreasingfamily { ε k } k and for each ε k one of the corresponding numbers n k = m ( ε k ),we obtain | a n k − q | ≤ ε k + | a n k − a n k − | → . The second part of the assertion is immediately satisfied. (cid:3)
Lemma 7 will help us to establish some bounded distortion properties.The following result implies in particular that every point x with χ ( x ) > Lemma 8.
Assume that J does not contain any critical points of f . Let x ∈ J be a point with χ ( x ) > . Then there exists a number K > such thatfor every q ∈ [ χ ( x ) , χ ( x )] \ { } there exists a number δ > and a sequence n k → ∞ such that i) lim k →∞ n k log | ( f n k ) (cid:48) ( x ) | = q , ii) Dist f n k | f − nkx ( B ( f nk ( x ) ,δ )) < K. Here K is a universal constant, while δ depends on the number q but not onthe point x .Proof. As J does not contain any critical point, the only accumulation pointsin J of the orbit of some critical point can be parabolic points. Let r be theleast common multiplier of the periods of all parabolic points in J . N THE LYAPUNOV SPECTRUM FOR RATIONAL MAPS 17
Given a number q ∈ [ χ ( x ) , χ ( x )] \ { } , there exists a number δ > y ∈ J is δ -close to some parabolic point then1 r log | ( f r ) (cid:48) ( y ) | ≤ q . Further, there exists a number δ > z ∈ J is δ -away from anyparabolic point then no orbit of a critical point passes through B ( z, δ ).To prove the claimed property, it will suffice to find a sequence { n k } k for which i) is satisfied and for which f n k ( x ) is in distance at least δ fromany parabolic point. Indeed, in such a situation the backward branch of f − n k x ( B ( f n k ( x ) , δ )) will not catch any critical point and the distortionestimations ii) will follow from the Koebe distortion lemma.If x is a Lyapunov regular point and q = lim n →∞ n log | ( f n ) (cid:48) ( x ) | is itsLyapunov exponent (which, by our assumptions, must be positive) then theclaim i) is automatically satisfied. In this case we just need to choose n k such that f n k ( x ) is far away from any parabolic point. Note that there mustbe infinitely many such times n k because otherwise the Lyapunov exponentat x would be no greater than q/ x is not a Lyapunov regular point, then we apply Lemma 7 to thesequence a n = 1 n log | ( f n ) (cid:48) ( x ) | . Notice that lim n →∞ ( a n +1 − a n ) = 0 is satisfied, because there are no criticalpoints in J and hence | f (cid:48) | is uniformly bounded. Hence, from the firstassertion of Lemma 7 we obtain a sequence { n k } k that satisfies i). Noticethat we have a n k + r = n k n k + r a n k + 1 n k + r log | ( f r ) (cid:48) ( f n k ( x )) | ≤ n k n k + r a n k + rn k + r q f n k ( x ) is δ -close to some parabolic point. This inequality cannotbe true for big n k (when a n k is already close to q ) because of the second partof assertion of Lemma 7. This proves that for any time n k large enough thepoint f n k ( x ) is δ -away from any parabolic point. (cid:3) We are now prepared to prove an upper bound for the Hausdorff dimen-sion of the level sets under consideration. To start with the most generalapproach that will be needed in the subsequent analysis, we first study aset of points x for which χ ( x ) > α, β ]. Let us first introduce some notation.Given 0 ≤ α ≤ β , β >
0, let(19) (cid:98) L ( α, β ) def = { x ∈ J : χ ( x ) ≤ β, χ ( x ) ≥ α, χ ( x ) > } . Proposition 2.
Assume that J does not contain any critical points of f .For every β > and ≤ α ≤ β we have dim H (cid:98) L ( α, β ) ≤ max α ≤ q ≤ β F ( q ) . If α > α + or β < α − then (cid:98) L ( α, β ) = ∅ .Proof. By Lemma 8, for every point x ∈ (cid:98) L ( α, β ) there exist a number q = q ( x ) ∈ [ α, β ] \ { } , a number δ >
0, and a sequence { n k } k of numbers suchthat(20) lim k →∞ n k log | ( f n k ) (cid:48) ( x ) | = q. and(21) 2 δ | ( f n k ) (cid:48) ( x ) | − K − ≤ diam f − n k x ( B ( f n k ( x ) , δ )) ≤ δ | ( f n k ) (cid:48) ( x ) | − K. Recall that for every d ∈ R there exists a exp ( P ( ϕ d ) − ϕ d )-conformalmeasure ν d that gives positive measure to any open set (see Section 2.4).Hence there exists c = c ( δ ) > n k we have c ≤ ν d ( B ( f n k ( x ) , δ )) ≤
1. Using again the distortion estimates, we can con-clude that(22) cK − d e − n k P ( ϕ d ) | ( f n k ) (cid:48) ( x ) | − d ≤ ν d ( f − n k x ( B ( f n k ( x ) , δ ))) ≤ K d e − n k P ( ϕ d ) | ( f n k ) (cid:48) ( x ) | − d , which implies thatlim k →∞ n k log ν d ( f − n k x ( B ( f n k ( x ) , δ ))) = − P ( ϕ d ) − d q and in particular that this limit exists. Hence, there exists N > n k ≥ N we have ν d (cid:0) f − n k x ( B ( f n k ( x ) ,δ )) (cid:1) ≥ e − n k ( P ( ϕ d )+ dq + dδ ) ≥ e − n k P ( ϕ d ) | ( f n k ) (cid:48) ( x ) | − d ≥ K − d e − n k P ( ϕ d ) (cid:0) diam f − n k x ( B ( f n k ( x ) , δ )) (cid:1) d (cid:18) δ (cid:19) d . Here we used Lemma 8 to obtain the last inequality. Applying (20) and (21)we yield(23) d ν d ( x ) ≤ P ( ϕ d ) q + d. For any q ∈ [ α, β ], q >
0, and ε > q , q ), q >
0, containing q and a number d ∈ R such that for all q ∈ ( q , q )(24) 1 q P ( ϕ d ) + d < (cid:40) F ( q ) + ε if F ( q ) (cid:54) = −∞ , −
100 if F ( q ) = −∞ . We can choose a countable family of intervals { ( q ( i )1 , q ( i )2 ) } i covering [ α, β ] \{ } and a sequence of corresponding numbers { d i } i . Defining the measure ν def = (cid:88) i − i ν d i N THE LYAPUNOV SPECTRUM FOR RATIONAL MAPS 19 we obtain d ν ( x ) ≤ inf i ≥ d ν di ( x ) ≤ max α ≤ q ≤ β F ( q ) + ε, where the second inequality follows from (24). Applying the Frostmanlemma we obtain thatdim H (cid:98) L ( α, β ) ≤ max α ≤ q ≤ β F ( q ) + ε Since ε can be chosen arbitrarily small, this finishes the proof of the firstclaim. The second claim was already proved in Lemma 5. (cid:3) Note that for every q ∈ [ α, β ], q >
0, we have L ( α, β ) ⊂ (cid:98) L ( q, q ), whichreadily proves the following result. Corollary 1.
Under the hypotheses of Proposition 2, we have dim H L ( α, β ) ≤ min α ≤ q ≤ β F ( q ) = min { F ( α ) , F ( β ) } . In particular, for every α > , we have dim H L ( α ) ≤ F ( α ) . The general case.
We now consider the general case that there arecritical points inside the Julia set.We need two technical results from the literature. The first one is thefollowing telescope lemma from [16]. Recall the definition of hyperbolictimes given in Section 2.5.
Lemma 9.
Given ε > and σ > , there exist constants K > and R > such that the following is true. Given x ∈ J with upper Lyapunov exponent χ ( x ) > σ , for every number r < R , for every n ≥ being a hyperbolic timefor x with exponent σ , and for every ≤ k ≤ n − we have diam f − n + kf k ( x ) ( B ( f n ( x ) , r )) ≤ r K e − ( n − k )( σ − ε ) . To formulate our second preliminary technical result we need the followingconstruction, see [9, 20, 21].
Pullback construction:
Fix some n > y ∈ J \ (cid:83) ni =1 f i (Crit). Fixsome R > { y i } ni =1 be some backward trajectory of y , i.e. y = y and y i +1 ∈ f − ( y i ) for every i = 1, . . . , n −
1. Let k be the smallest integerfor which f − k y k ( B ( y, R )) contains a critical point. For every (cid:96) ≥ k (cid:96) +1 be the smallest integer greater than k (cid:96) such that f − ( k (cid:96) +1 − k (cid:96) ) y k(cid:96) +1 ( B ( y k (cid:96) , R ))contains a critical point and so on. In this way, for each backward branch { y i } i we construct a sequence { k (cid:96) } (cid:96) that must have a maximal element notgreater than n . Let this element be k and consider the set Z of all pairs( y k , k ) built from all the backward branches of f that start from y . Let N ( y, n, R ) def = Z . We have the following estimate, see [21, Lemma 3.7]and [20, Appendix A]. Lemma 10.
Given ε > , there exist K > and R > such that for all R ≤ R we have N ( y, n, R ) < K e nε uniformly in y and n . Recall the definition of conical points in Section 2.5. Using the above twolemmas we can now show the following result.
Proposition 3.
The set of points x ∈ J that are not conical and satisfy χ ( x ) > has Hausdorff dimension zero.Proof. Let us choose some numbers σ > ε >
0. Let r def = 12 K + 4 min { R , R } , where K and R are constants given by Lemma 9 and where R is givenby Lemma 10.We can choose a finite family of balls { B i } Li =1 of radius 3 r such thatany ball of radius 2 r intersecting J must be contained in one of the balls B i . In the case that we can prove existence of a sequence { n i } i such that f − ( n i − k ) f k ( x ) ( B ( f n i ( x ) , r )) does not contain critical points for any 0 ≤ k ≤ n i −
1, the Koebe distortion lemma will imply that x is a conical point for r , { n i } i , and U i = f − n i x ( B ( f n i ( x ) , r )).Let G ( m, σ ) be the set of points x ∈ J with upper Lyapunov exponentgreater than σ for which with r chosen above for all n > m the backwardbranch of f − n from ball B ( f n ( x ) , r ) onto a neighborhood of x will neces-sarily meet a critical point, that is, for some 0 ≤ k ≤ n − f − ( n − k ) f k ( x ) ( B ( f n ( x ) , r )) ∩ Crit (cid:54) = ∅ . First we claim that dim H G ( m, σ ) = 0. Let us denote by G ( m, σ, n ) thesubset of G ( m, σ ) for which n > m is a hyperbolic time with exponent σ .Recall that χ ( x ) > σ > x with exponent σ . Hence, we have(25) G ( m, σ ) = (cid:92) m ≥ m (cid:91) n>m G ( m, σ, n ) . Let x ∈ G ( m, σ, n ). Let B j = B ( y, r ) be the ball that contains B ( f n ( x ) , r ).We will apply the “pullback construction” for Lemma 10 to the point y , thenumbers n , R = min { R , R } , and to the backward branch f − nx . Andlet k = max (cid:96) k (cid:96) and y k be given by the pullback construction (compareFigure 1).We first note that Lemma 9 and ρ ( f n ( x ) , y ) ≤ r imply that ρ ( f n − k ( x ) , y k ) ≤ r K . This implies B ( f n − k ( x ) , r ) ⊂ B ( y k , R ) and hence k ≥ n − m because x ∈ G ( m, σ ). Thus, with fixed n and B j , by Lemma 10 the point x must belongto one of at most K e nε (deg f ) m preimages of B j . As B ( y, r ) ⊂ B ( x, r ) N THE LYAPUNOV SPECTRUM FOR RATIONAL MAPS 21 ≤ m y k2 y k1 y k ≤ (deg f) m xy n y f (x) n B j n d e p t h o f i t e r a t i o n f (B ) x-n j branches Figure 1.
Pullback construction starting from the point y .and 4 r < R , by Lemma 9 this pre-image of B j has diameter not greaterthan 8 r K e − n ( σ − ε ) .Hence we showed that every point in G ( m, σ, n ) belongs to the n th pre-image of some ball B j along a backward branch for that k ≥ m − n (where k = max (cid:96) k (cid:96) is as in the pullback construction). Thus, by Lemma 9 the set G ( m, σ, n ) is contained in a union of at most LK e nε (deg f ) m sets of diam-eter not greater than 8 r K e − n ( σ − ε ) . Using those sets to cover G ( m, σ, n )and applying (25), we obtaindim H G ( m, σ ) ≤ εσ − ε . As ε can be chosen arbitrarily small, the claim follows.Finally note that the set of points that we want to estimate in the propo-sition is contained in the union ∞ (cid:91) m =1 ∞ (cid:91) n =1 G (cid:0) m, n (cid:1) . Since for any set in this union its Hausdorff dimension is zero, the assertionfollows. (cid:3)
Note that in the above proof we were able to show something more.Namely notice that the choice of r depended on σ alone (and not directlyon the point x ).We are now prepared to prove the following estimate. Proposition 4.
Let < α ≤ β ≤ α + . We have dim H L ( α, β ) ≤ max (cid:26) , max α ≤ q ≤ β F ( q ) (cid:27) . Proof.
Let α and β be like in the assumptions and let x ∈ L ( α, β ). ByProposition 3 we can restrict our considerations to the case that x is aconical point with corresponding number r >
0, sequence { n k } k , and familyof neighborhoods { U k } k . Hence, there exist numbers δ > K > δ | ( f n k ) (cid:48) ( x ) | − K − ≤ diam f − n k x ( B ( f n k ( x ) , δ )) ≤ δ | ( f n k ) (cid:48) ( x ) | − K. Choosing, if necessary, a subsequence of { n k } k , we can find q = q ( x ) ∈ [ α, β ]for which we have(27) lim k →∞ n k log | ( f n k ) (cid:48) ( x ) | = q. Recall that for any d ∈ R there exists a exp ( P ( ϕ d ) − ϕ d )-conformal mea-sure ν d that gives positive measure to any open set that intersects J (seeSection 2.4). Hence there exists a number c δ > n k wehave c δ ≤ ν d ( B ( f n k ( x ) , δ )) ≤
1. Using again distortion estimates, we canconclude that(28) c e − n k P ( ϕ d ) K − | ( f n k ) (cid:48) ( x ) | − d ≤ ν d ( f − n k x ( B ( f n k ( x ) , δ ))) ≤ e − n k P ( ϕ d ) K | ( f n k ) (cid:48) ( x ) | − d . The rest of the proof is similar to the proof of Proposition 2. First we obtainthat for every d ∈ R we have d ν d ( x ) ≤ P ( ϕ d ) q + d. Choosing the right number d , we then conclude that d ν d ( x ) ≤ F ( q ). Wefinish the proof by constructing a measure such that for an arbitrarily smallchosen number ε and some number q ∈ [ α, β ] the lower pointwise dimensionof that measure at every x ∈ L ( α, β ) is not greater than F ( q ) + ε . (cid:3) Remark 3.
Notice that in the general case, in which we do have criticalpoints in J , for a point x ∈ J with χ ( x ) < χ ( x ) we cannot apply thesame techniques as in Section 4.1. In particular, in the above proof foreach point x ∈ L ( α, β ) we only know that there exists some number q = q ( x ) ∈ [ χ ( x ) , χ ( x )] to which the Lyapunov exponents over some subsequenceof times { n k } k will converge, while in Proposition 2 we were able to take anarbitrary number q in that interval. Hence, to show the following result wecan only consider the set L ( α ) = L ( α, α ) and not (cid:98) L ( q, q ) for an arbitrary N THE LYAPUNOV SPECTRUM FOR RATIONAL MAPS 23 number q ∈ [ α, β ], and the following corollary is weaker than Corollary 1.However, the implications for the regular part of the spectrum remain thesame.Proposition 4 and L ( α ) = L ( α, α ) readily imply the following result. Corollary 2.
For α ∈ [ α − , α + ] \ { } we have dim H L ( α ) ≤ F ( α ) . Lower bounds for the dimension
In this section we will derive lower bounds for the Hausdorff dimension.We will either assume that f is a non-exceptional map or that f is excep-tional but Σ ∩ J = ∅ . Recall that under those assumptions Proposition 1 isvalid, so we can approximate the pressure with respect to f | J with pressuresthat are defined with respect to a sequence of Cantor repellers f a m | Λ m thatwere constructed in Section 2.3.One more case we would like to exclude is α − = α + = α . It is not veryinteresting because in this case any measure supported on any hyperbolicset Λ m ⊂ J has Lyapunov exponent α . Hence we automatically havedim H L ( α ) ≥ sup m ≥ F m ( α )and the supremum on the right hand side is in this case equal to F ( α ).Therefore, in the following considerations we will assume that α − < α + .5.1. The interior of the spectrum.
We use the sequence of Cantor re-pellers to obtain for any exponent from the interior of the spectrum a biguniformly expanding subset of points with Lyapunov exponent α that pro-vides us with an estimate from below. Proposition 5.
For α ∈ ( α − , α + ) we have dim H L ( α ) ≥ F ( α ) .Proof. Let us consider the sequence of Cantor repellers f a m | Λ m from Propo-sition 1. By (11), for each number α + > α > α − there exists m ≥ α + m > α > α − m for every m ≥ m . Obviously we havedim H L ( α ) ≥ sup m ≥ dim H L ( α ) ∩ Λ m . Since Λ m is a uniformly expanding repeller with respect to f a m , for anyexponent α ∈ ( α − m , α + m ) there exists a unique number q = q ( α ) ∈ R suchthat α = − a m dds P f am | Λ m ( ϕ s ) | s = q ( α ) and an equilibrium state µ q for thepotential ϕ q (with respect to f a m | Λ m ) such that the Lyapunov exponent of µ q with respect to f a m is equal to a m α (compare the classical results in theintroduction). For the measure ν m = a m − (cid:88) i =0 µ m ◦ f i we have χ ( ν m ) = α . Hence, the variational principle impliesmax (cid:110) h ν ( f ) : ν ∈ M E (cid:16) (cid:91) i f i (Λ m ) (cid:17) , χ ( ν ) = α (cid:111) ≥ a m P f am | Λ m ( S a m ϕ q ) + qα ≥ inf d ∈ R (cid:18) a m P f am | Λ m ( S a m ϕ d ) + dα (cid:19) = F m ( α ) . We obtain that dim H ν = h ν ( f ) χ ( ν ) whenever ν is an f -invariant ergodic Borelprobability measure with positive Lyapunov exponent [11]. This impliesthat for every m ≥ m dim H L ( α ) ∩ (cid:91) i f i (Λ m ) ≥ max (cid:110) h ν ( f ) α : ν ∈ M E (cid:16) (cid:91) i f i (Λ m ) (cid:17) , χ ( ν ) = α (cid:111) . From here the we can conclude that dim H L ( α ) ≥ sup m ≥ F m ( α ). Togetherwith Proposition 1 the statement is proved. (cid:3) The boundary of the spectrum.
Unfortunately, the above approachdoes not suffice to analyze the level sets for exponents from the boundary ofthe spectrum. Our main goal in this section is to prove the following result.It will not only enable us to describe the boundary of the spectrum but alsoprovide us with dimension lower bounds for level sets of irregular points.
Theorem 3.
Let { Λ i } i be a sequence of subsets of J . We assume that each Λ i is a uniformly expanding repeller for some iteration f a i and contains non-immediately postcritical points. Let { φ i } i be a sequence of H¨older continuouspotentials and let { µ i } i be a sequence of equilibrium states for φ i with respectto f a i | Λ i . Then dim H (cid:26) x ∈ J : χ ( x ) = lim inf i →∞ χ ( µ i ) , χ ( x ) = lim sup i →∞ χ ( µ i ) (cid:27) ≥ lim inf i →∞ dim H µ i and dim P (cid:26) x ∈ J : χ ( x ) = lim inf i →∞ χ ( µ i ) , χ ( x ) = lim sup i →∞ χ ( µ i ) (cid:27) ≥ lim sup i →∞ dim H µ i . We derive the following estimates for level sets that include exponents atthe boundary of the spectrum.
Proposition 6.
For α − ≤ α < β ≤ α + we have dim H (cid:98) L ( α, β ) ≥ max α ≤ q ≤ β F ( q ) . Proof.
The claimed estimate follows from Theorem 3.We can also observe that for every q ∈ ( α, β ) we have (cid:98) L ( α, β ) ⊃ L ( q, q ) = L ( q ). Hence we can apply Proposition 5 to derive dim H L ( q ) ≥ F ( q ) andprove the claimed estimate. (cid:3) N THE LYAPUNOV SPECTRUM FOR RATIONAL MAPS 25
Proposition 7.
For α − ≤ α ≤ β ≤ α + we have dim H L ( α, β ) ≥ min { F ( α ) , F ( β ) } . Proof.
Since we assume α − < α + , there exists a sequence { Λ i } i of uniformlyexpanding repellers and a sequence { µ i } i of equilibrium states for the po-tential − log | f (cid:48) | with respect to f | Λ i such that dim H µ i = F ( χ ( µ i )) and α = lim i →∞ χ ( µ i ) and β = lim i →∞ χ ( µ i +1 ). Theorem 3 then implies that L ( α, β ) = (cid:8) x ∈ J : χ ( x ) = α, χ ( x ) = β (cid:9) ≥ lim inf i →∞ F ( χ ( µ i )) . Since lim inf i →∞ F ( χ ( µ i )) ≥ min { F ( α ) , F ( β ) } , this proves the claimed esti-mate. (cid:3) Recall that a point x is said to be recurrent if f n i ( x ) → x for somesequence n i (cid:37) ∞ . Corollary 3.
Assume that J does not contain any recurrent critical pointsof f , that f is non-exceptional, and that F (0) (cid:54) = −∞ . Then we have dim H L (0) = dim H J = F (0) . Proof.
Proposition 7 implies that dim H L (0) ≥ F (0). As F (0) (cid:54) = −∞ , thepressure function d (cid:55)→ P ( ϕ d ) is nonnegative and F (0) = inf { d : P ( ϕ d ) = 0 } .Since J does not contain any recurrent critical points of f , we can apply [24,Theorem 4.5] and conclude that inf { d : P ( ϕ d ) = 0 } = dim H J . (cid:3) To prove Theorem 3, we will construct a sufficiently large set of pointsthat have precisely the given lower/upper Lyapunov exponent. Note thatsuch a set will not “be seen” by any invariant measure in the non-trivialcase that lower and upper exponents do not coincide. Also points withLyapunov exponent zero, though Lyapunov regular, will not “be seen” byany interesting invariant measure. Our approach is to show that such a setis large with respect to some necessarily non-invariant probability measure.It is generalizing the construction of so-called w-measures introduced in [8]for which we strongly made use of the Markov structure that is not availableto us here.
A technical lemma.
In preparation for the proof of Theorem 3 we start withsome technical result that will be usefull shortly after.
Lemma 11.
Let g : C → C be a conformal map and Λ ⊂ C be a compact g -invariant hyperbolic topologically transitive set, µ a g -invariant ergodicmeasure on Λ with Lyapunov exponent χ = χ ( µ ) , entropy h = h µ ( g ) , andHausdorff dimension d = d ( µ ) . Let V be an open set of positive measure µ .If γ is small enough then for any ε > for µ -almost every point v ∈ Λ thereexist a number K > and a sequence { n i } i such that for each n i there is aset F n i ⊂ V ∩ Λ such that for all y j ∈ F n i i) g n i ( y j ) = v , ii) K − exp( m ( χ − ε )) < | ( g m ) (cid:48) ( y j ) | < K exp( m ( χ + ε )) for all m ≤ n i , iii) the branch g − n i y j mapping v onto y j extends to all B ( v, γ ) and thedistortion of the resulting map is bounded by K , iv) we have µ (cid:91) y j ∈ F ni g − n i y j ( B ( v, γ )) ≥ K − , v) for j (cid:54) = k we have ρ (cid:16) g − n i y j ( B ( v, γ )) , g − n i y k ( B ( v, γ )) (cid:17) > K − diam g − n i y j ( B ( v, γ )) , vi) for any x ∈ V and r > we have µ B ( x, r ) ∩ (cid:91) y j ∈ F ni g − n i y j ( B ( v, γ )) ≤ Kr d − ε , vii) we have K − e − n ( h + ε ) ≤ µ (cid:16) g − n i y j ( B ( v, γ )) (cid:17) ≤ e − n ( h − ε ) . Proof. As g | Λ is uniformly expanding, there exist numbers γ > c > n ≥ y ∈ Λ, v = f n ( y ) ∈ Λ, and a backward branch g − ny mapping v onto y for every x , x ∈ B ( v, γ ) we have(29) | ( g − ny ) (cid:48) ( x ) || ( g − ny ) (cid:48) ( x ) | ≤ c , and in particular the mapping g − ny extends to all of B ( v, γ ). Moreover, thereis some constant c > n or y such thatdiam g − ny ( B ( v, γ )) ≤ c γ. We assume that γ is so small that for any two points x , y with distance fromΛ and mutual distance < c γ , for any point x (cid:48) ∈ g − ( x ) there is at mostone point y (cid:48) ∈ g − ( y ) such that ρ ( x (cid:48) , y (cid:48) ) ≤ c γ . This is true for any smallenough number γ because Λ is in positive distance from any critical pointof g . Let (cid:101) V ⊂ V be such that B ( (cid:101) V , c γ ) ⊂ V and that (cid:101) V is nonempty andof positive measure, which is possible whenever γ is small enough.Notice that B ( v, c +1 γ ) has positive µ -measure for µ -almost every v .Choose such point v and let (cid:101) U def = B (cid:18) v, c + 1 γ (cid:19) , U def = B ( v, γ ) . Let δ def = 12 min (cid:110) µ ( (cid:101) U ) , µ ( (cid:101) V ) (cid:111) . Let ε >
0. There is a set of points Λ (cid:48) ⊂ Λ with µ (Λ (cid:48) ) > − δ (actually, itcan be chosen to have arbitrarily large measure) and a number N > n ≥ N and for every x ∈ Λ (cid:48) we have N THE LYAPUNOV SPECTRUM FOR RATIONAL MAPS 27 (C1) (cid:12)(cid:12)(cid:12)(cid:12) n (cid:110) k ∈ { , . . . , n − } : g k ( x ) ∈ (cid:101) U ∩ Λ (cid:111) − µ ( (cid:101) U ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε ,(C2) (cid:12)(cid:12) log | ( g n ) (cid:48) ( x ) | − nχ (cid:12)(cid:12) ≤ nε, (C3) | log Jac µ g n ( x ) − nh | ≤ nε ,(C4) for r ≤ diam Λ we have µ ( B ( x, r )) < c r d − ε . Here (C1), (C2) simply follow from ergodicity, (C3) is a consequence of theRokhlin formula and ergodicity, and (C4) follows from the definition of thedimension d ( µ ).We choose now a family { E n } n ≥ N of subsets of Λ (cid:48) such that for every n ≥ N the set E n is a maximal ( n, c c +1 γ )-separated subset of Λ (cid:48) . Given n ≥ N , let V n def = E n ∩ (cid:101) V . For each n ≥ N let F n def = (cid:8) x ∈ V n : g − nx ( U ) ⊂ V (cid:9) . For every z j ∈ F n let V n,j def = g − nz j ( (cid:101) U ). Obviously, all the sets V n,j are pairwisedisjoint.By (C1) the trajectory of every point from (cid:101) V ∩ Λ (cid:48) visits (cid:101) U at some time n ≥ N (in fact, at infinitely many times). Let x be such a point and n be such a time, that is, g n ( x ) ∈ (cid:101) U . Because E n is maximal, x must be( n, c c +1 γ )-close to some point z j ∈ E n . Hence we have g n ( z j ) ∈ B (cid:18) g n ( x ) , c c + 1 γ (cid:19) ⊂ U and x ∈ V n,j ⊂ g − nz j ( U ) ⊂ B ( x, c γ ) ⊂ V. This shows that z j ∈ F n . This together with (C1) implies that n − (cid:88) (cid:96) =0 µ (cid:91) j V (cid:96),j ≥ ( µ ( (cid:101) U ) − ε ) n µ (Λ (cid:48) )and hence, as it is satisfied for all n > N , we obtain (cid:88) j µ ( V (cid:96),j ) ≥
12 ( µ ( (cid:101) U ) − ε ) µ (Λ (cid:48) )for infinitely many (cid:96) ≥ N . This gives us a sequence of times n i def = (cid:96) andpoints y j = g − n i z j ( v ) for which the assertion of the lemma is true. Indeed, g − n i z j ( U ) and g − n i z k ( U ) for j (cid:54) = k are disjoint because we are in a sufficientlylarge distance from Crit. We also know that those maps g − n i z j | U have uni-formly bounded distortion. Recall that U = B ( v, γ ), which implies that g − n i z j ( B ( v, γ/ g − n i z k ( B ( v, γ/ j (cid:54) = k are not only disjoint but indistance that is comparable to the sum of their diameters and v) follows. Λ Sy x f v U ˷ US ˷ j x-n f n Figure 2.
Finding y j Further properties i) and iii) are checked directly from the construction.Moreover, vi) follows from (C4), iv) follows from our choices of (cid:96) , ii) from(C2) and vii) from iv) and (C3). (cid:3)
Construction of a Cantor set.
We now continue with some preliminary con-structions that will be needed in the proof of Theorem 3.As a first step, we are going to construct “bridges” between the repellersΛ i (compare Figure 3). This is very similar to the proof of Lemma 2 thoughhere we will not necessarily require that the repellers are disjoint. To fixsome notation, let { ( B i , b i ) } i be a collection of bridges , that is, let B i def = B ( z i , r i ) ⊂ B (Λ i , γ i ) , where the numbers r i , b i , γ i and the points z i are appropriately chosen suchthat f b i | B i is a homeomorphism and that µ i +1 ( f b i ( B i )) > , f b i ( z i ) ∈ Λ i +1 , and ρ ( f k ( B i ) , Crit) ≥ δ i for every 1 ≤ k ≤ b i . The particular choice of the numbers and points will be specified in thefollowing so that we are able to apply Lemma 11 to the each of the sets Λ i and the maps g = f a i .Let us outline the following Cantor set construction. Lemma 11 enablesus to select sufficiently many preimages for each of the disks B i (as wechoose B i ⊂ B ( v i , γ i ) and then select preimages g − n i ( v i ) using the lemma).The construction of this Cantor set is easily described in terms of backwardbranches: at level i we start with the disk B i , apply Lemma 11 and finda large number of components of f − a i n i ( B i ) in f b i − ( B i − ), then we go N THE LYAPUNOV SPECTRUM FOR RATIONAL MAPS 29 backwards through the bridge obtaining the components in B i − . We repeatthe procedure in B i − , . . . , B .Given the sequence of uniformly expanding repellers f a i | Λ i with equilib-rium states µ i with Lyapunov exponent χ ( µ i ) and entropy h µ i ( f a i ) (bothwith respect to the map f a i ), let us denote χ i def = 1 a i χ ( µ i ) , h i def = 1 a i h µ i ( f a i ) , d i def = d ( µ i ) = h i χ i , and s i def = | ( f b i ) (cid:48) ( z i ) | , t i def = Dist f b i | B i . Let w i def = inf x : ρ ( x, Crit) >δ i | f (cid:48) ( x ) | , W def = sup x ∈ J | f (cid:48) ( x ) | . The constants γ i can be chosen to be arbitrarily small (at the cost of decreas-ing r i and increasing b i , thus changing t i and s i accordingly). In particularwe can choose each γ i sufficiently small such that Lemma 11 applies to themap g = f a i and the set Λ i . Let V def = B (Λ , γ ) and V i +1 def = f b i ( B i ) . We choose a fast decreasing sequence { ε i } i . We will denote by K i thenumbers K i = K (Λ i , µ i , V i , ε i , v i ) as given by Lemma 11 (for g = f a i ) wherewe choose each point v i ∈ Λ i such that B i ⊂ B ( v i , γ i ). Let A def = f − a n y j ( B )for one point y j and n as provided by Lemma 11. We have Dist f a n | A ≤ K and for every x ∈ A and m ≤ n we have K − e a m ( χ − ε ) < | ( f a m ) (cid:48) ( x ) | < K e a m ( χ + ε ) (note that n can be chosen to be arbitrarily big, as guaranteed by Lemma 11).We apply Lemma 11 now to Λ , µ , V , ε , v , g = f a , which pro-vides us with a family F ∈ f − a n ( v ). Let (cid:98) A j = f − a n y j ( B ( v , γ )) for y j ∈ F . Those sets are contained in the set V = f b ( B ). We also haveDist f a n | (cid:98) A j ≤ K and for every x ∈ (cid:98) A j and m ≤ n we have K − e a m ( χ − ε ) < | ( f a m ) (cid:48) ( x ) | < K e a m ( χ + ε ) . Such sets will be used later to distribute the w-measure accordingly. Itprovides us also with a family of sets (cid:101) A j ⊂ (cid:98) A j such that f a n ( (cid:101) A j ) = B .Such sets will be used later to define our Cantor set on which the w-measureis supported.We repeat this procedure for every i , using Lemma 11 repeatedly: for ev-ery i > F i ⊂ f − a i n i ( v i ) and the corresponding components (cid:98) A ij of f − a i n i ( B ( v i , γ i )) contained in V i that satisfy ρ ( (cid:98) A ij , (cid:98) A ik ) ≥ K − i diam (cid:98) A ij , j (cid:54) = k, and for every x ∈ (cid:98) A ij and m ≤ n i K − i e a i m ( χ i − ε i ) < | ( f a i m ) (cid:48) ( x ) | < K i e a i m ( χ i + ε i )0 KATRIN GELFERT, FELIKS PRZYTYCKI, AND MICHA(cid:32)L RAMS
28 KATRIN GELFERT, FELIKS PRZYTYCKI, AND MICHA ! L RAMS Λ Λ Λ A S S f n f n f n f b f b G G G v v v Figure 3.
Connecting the hyperbolic sets by bridges. B B B
328 KATRIN GELFERT, FELIKS PRZYTYCKI, AND MICHA ! L RAMS Λ Λ Λ A S S f n f n f n f b f b G G G v v v Figure 3.
Connecting the hyperbolic sets by bridges. B B B
328 KATRIN GELFERT, FELIKS PRZYTYCKI, AND MICHA ! L RAMS Λ Λ Λ A S S f n f n f n f b f b G G G v v v Figure 3.
Connecting the hyperbolic sets by bridges. B B B A B = B(z ,r ) Sf v
B B S n f a n f a n f b f b v v Λ Λ Λ Figure 3.
Connecting the hyperbolic sets by bridges.and Dist f a i n i | (cid:98) A ij ≤ K i . In addition, by Lemma 11 iv) we have µ i (cid:91) j (cid:98) A ij ≥ K − i , and by Lemma 11 vi) for any x ∈ V i and r > µ i B ( x, r ) ∩ (cid:91) j (cid:98) A ij ≤ K i r d i − ε i . Note here that for every i ≥ r i s i t − i ≤ diam V i +1 ≤ r i s i t i . Let m i def = i (cid:88) k =1 ( a k n k + b k ) . Let A j be the component of A ∩ f − m ( (cid:101) A j ) for which f a n ( A j ) ⊂ B .Similarly, let A i j ...j i − be the component of A ( i − j ...j i − ∩ f − m i − ( (cid:101) A i j i − )for which f a i − n i − + m i − ( A i j ...j i − ) ⊂ B i − .The sets A i def = (cid:91) j ...j i − A i j ...j i − form a decreasing sequence of unions of topological balls. Moreover, the pair (cid:0) A ( i − j ...j i − , { A i j ...j i − k } k (cid:1) is an image of ( V i , { (cid:101) A i k } k ) under a branch ofthe map f − m i − , the distortion of that branch is bounded by (cid:101) K i − def = i − (cid:89) k =1 K k t k , N THE LYAPUNOV SPECTRUM FOR RATIONAL MAPS 31 A AA AA
211 31 ˷ ˷
S S
A A
311 3kl ff m -m m A ˷ A A ˷ Figure 4.
Local structure of the Cantor set A and the absolute value of its derivative is between(30) L i − def = i − (cid:89) k =1 s − k t − k e − a k n k ( χ k + ε k ) and (cid:98) L i − def = i − (cid:89) k =1 s − k t k e − a k n k ( χ k − ε k ) ( (cid:101) K i , L i , (cid:98) L i depend on i only). We can now define the Cantor set(31) A def = (cid:92) i ≥ A i (compare Figure 4). We will summarize its geometric and dynamic proper-ties in the following lemma. Lemma 12.
The above defined set A possesses the following properties: i) (Lyapunov exponents on the islands) for x ∈ A i and k ≤ n i we have K − i e a i k ( χ i − ε i ) < | ( f a i k ) (cid:48) ( f m i − ( x )) | < K i e a i k ( χ i + ε i ) , ii) (Lyapunov exponents on the bridges) for x ∈ A i and k ≤ b i we have w ki < | ( f k ) (cid:48) ( f m i − + a i n i ( x )) | < W k , iii) we have K − i (cid:101) K − i − L i − r i e − a i n i ( χ i + ε i ) ≤ diam A ij ...j i − ≤ K i (cid:101) K i − (cid:98) L i − r i e − a i n i ( χ i − ε i ) , iv) there are at least exp( a i n i ( h i − ε i )) sets A i j ...j i − contained in everyset A ( i − j ...j i − , v) for any k i − (cid:54) = j i − and any i, j , . . . , j i − we have ρ ( A ij ...j i − , A ij ...j i − k i − ) ≥ (cid:101) K − i − K − i diam A ij ...j i − vi) we have µ i (cid:91) j (cid:98) A ij ≥ K − i , vii) for any x ∈ V i and r > we have µ i B ( x, r ) ∩ (cid:91) j (cid:98) A ij ≤ K i r d i − ε i , viii) we have K − i e − a i n i ( h i + ε i ) ≤ µ i (cid:16) (cid:98) A ij (cid:17) ≤ e − a i n i ( h i − ε i ) . Additional assumptions on { n i } i and properties guaranteed by Lemma 11will enable us to estimate the Hausdorff and packing dimensions of theconstructed set and describe the upper and lower Lyapunov exponents ateach point. This will be done in the following. Construction of a w-measure on the Cantor set.
We continue to consider theCantor set A constructed in (31). Let µ be the probability measure whichon each level i is distributed on the cylinder sets A i j ...j i − of level i in thefollowing way(32) µ ( A i j ...j i − ) def = µ ( A i − j ...j i − ) µ i ( (cid:98) A i j i − ) (cid:80) k µ i ( (cid:98) A i k ) . We extend the measure µ arbitrarily to the Borel σ -algebra of A . We call theprobability measure a w-measure with respect to the sequence { f | Λ i , φ i , µ i } i .After these preparations, we are now able to prove Theorem 3. We willcontinue to use the notations in the above construction of the set A . Proof of Theorem 3.
In the course of the following proof we will choose somesequence { ε k } k and then construct a sequence of positive integers { n i } i . Hereeach of those numbers n i has to satisfy several conditions that depend on { ε k } k , the parameters of the hyperbolic sets { Λ k } k and the measures { µ k } k ,and the previously chosen numbers n j , j = 1, 2, . . . , i −
1. Naturally, it isalways possible to satisfy all those conditions at the same time.We will first check that the Cantor set defined in (31) satisfies A ⊂ L (cid:16) lim inf χ ( µ i ) , lim sup χ ( µ i ) (cid:17) (under some appropriate assumptions about { n i } i ). Then we will estimatethe Hausdorff and packing dimensions of A using the w-measure µ definedin (32).Let us first consider the Lyapunov exponent at a point in the set A . Let (cid:96) n ( x ) def = 1 n log | ( f n ) (cid:48) ( x ) | . N THE LYAPUNOV SPECTRUM FOR RATIONAL MAPS 33
For n ≤ a n we have (cid:96) n ( x ) ∈ (cid:18) χ − ε − n log K − O ( a n ) , χ + ε + 1 n log K + O ( a n ) (cid:19) . We know that f n + k ( x ) stays in distance at least δ from the critical pointsfor every k ≤ b . Hence, for a n < n ≤ a n + b we have(33) (cid:96) n ( x ) = a n n (cid:96) a n ( x ) + n − a n n O ( | log w | )and for n big enough (in comparison with log K and b | log w | ) the righthand side of (33) is between χ − ε and χ + 2 ε . For m i − < n ≤ m i − + a i n i we have (cid:96) n ( x ) = m i − n (cid:96) m i − ( x ) + 1 n log | ( f n − m i − ) (cid:48) ( f m i − ( x )) | + O ( a i n )while for m i − + a i n i < n ≤ m i we have (cid:96) n ( x ) = m i − n (cid:96) m i − ( x )+ 1 n log | ( f a i n i ) (cid:48) ( f m i − ( x )) | + n − m i − − a i n i n O ( | log w i | ) . Estimating the second summand using Lemma 12 i) and the third one usingLemma 12 ii) and assuming that n i is big enough (in comparison with n i − , b i | log w i | , a i +1 , and log K i ), we can first prove that | (cid:96) m i ( x ) − χ i | < ε i (by induction) and then prove that for all m i < n < m i +1 we have (cid:12)(cid:12)(cid:12)(cid:12) (cid:96) n ( x ) − (cid:18) m i n χ i + n − m i n χ i +1 (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) < ε i + ε i +1 ) . As the upper (the lower) Lyapunov exponent of x equals the upper (thelower) limit of (cid:96) n ( x ) as n → ∞ , we have shown that every point x ∈ A satisfies χ ( x ) = lim inf i →∞ χ ( µ i ) , χ ( x ) = lim sup i →∞ χ ( µ i ) . Let us now estimate the Hausdorff and packing dimensions of the set A .To do so, we will apply the Frostman lemma. Let us calculate the pointwisedimension of the measure µ defined in (32) at an arbitrary point x ∈ A .Notice that we can write { x } = ∞ (cid:92) i =1 A i j ...j i − for some appropriate symbolic sequence ( j j . . . ). By Lemma 12 v), theball B ( x, r ) does not intersect any of the sets A i k ...k i − if k . . . k i − (cid:54) = j . . . j i − whenever we have r ≤ R i ( x ) def = (cid:101) K − i − K − i diam A i j ...j i − . Consider R i +1 ( x ) < r ≤ R i ( x ). We have µ ( B ( x, r )) ≤ (cid:88) A i +1 j ...ji − k ∩ B ( x,r ) (cid:54) = ∅ µ ( A i +1 j ...j i − k ) ≤ (cid:88) A i +1 j ...ji − k ⊂ B (cid:16) x,r +max (cid:96) diam A i +1 j ...ji − (cid:96) (cid:17) µ ( A i +1 j ...j i − k ) . Let D i def = max (cid:96) diam A i +1 j ...j i − (cid:96) . We continue with µ ( B ( x, r )) ≤ µ ( A i j ...j i − ) (cid:80) (cid:96) µ i +1 ( (cid:98) A i +1 (cid:96) ) (cid:88) k : (cid:98) A i +1 k ⊂ f mi ( B ( x,r + D i )) µ i +1 ( (cid:98) A i +1 k ) ≤ µ ( A i j ...j i − ) (cid:80) (cid:96) µ i +1 ( (cid:98) A i +1 (cid:96) ) µ i +1 (cid:0) B (cid:0) f m i ( x ) , L − i ( r + D i ) (cid:1)(cid:1) . Using Lemma 12 vi) and vii) we can estimate µ ( B ( x, r )) ≤ µ ( A i j ...j i − ) K i +1 µ i +1 (cid:0) B (cid:0) f m i ( x ) , L − i ( r + D i ) (cid:1)(cid:1) ≤ µ ( A i j ...j i − ) K i +1 L d i +1 − ε i +1 i ( r + D i ) d i +1 − ε i +1 . (34)Let(35) Ξ def = µ ( A i j ...j i − ) K i +1 L d i +1 − ε i +1 i . Using Lemma 12 vi) and viii), we can now estimate the first factor in (35)and we obtainlog µ ( A i j ...j i − ) ≤ i (cid:88) k =2 log µ ( A k j ...j k − ) µ ( A k − j ...j k − ) ≤ i (cid:88) k =2 (cid:16) log µ k (cid:16) (cid:98) A k j k − (cid:17) + log K k (cid:17) ≤ − i (cid:88) k =2 ( a k n k ( h k − ε k ) + log K k ) ≤ − a i n i ( h i − ε i )provided that we assume that n i has been chosen big enough (in comparisonwith a k , n k , log K k , k < i ). For the second factor in (35) we yield − log L i = i (cid:88) k =1 log s k + log t k + a k n k ( χ k + ε k ) ≤ a i n i ( χ i + 2 ε i ) . provided that we assume that n i has been chosen big enough (in comparisonwith a k , n k , log K k , and s k , k < i ). This implies thatlog Ξ = a i n i χ i ( d i +1 − d i ) + n i O ( ε i , ε i +1 ) . N THE LYAPUNOV SPECTRUM FOR RATIONAL MAPS 35
Further, using (30) and Lemma 12 iii) we obtaindiam A i +1 j ...j i − (cid:96) R i +1 ≤ (cid:101) K i K i +1 i +1 (cid:89) k =1 e n k ε k and hence D i ≤ e n i +1 ε i +1 R i +1 . In addition, by Lemma 12 iii) we havelog R i +1 = a i +1 n i +1 ( χ i +1 + O ( ε i +1 ))and log D i ≤ − a i +1 n i +1 ( χ i +1 + O ( ε i +1 )) . assuming that n i has been chosen big enough. To estimate (34), we considernow the two cases: a) that r ≤ D i and b) that r > D i . In case a) we havethat µ ( B ( x, r )) ≤ Ξ (2 D i ) d i +1 − ε i +1 . Note that the the right hand side of this estimate no longer depends on r and hence,(36)log µ ( B ( x, r ))log r ≥ log (cid:0) Ξ · (2 D i ) d i +1 − ε i +1 (cid:1) log R i +1 = O (cid:18) n i n i +1 (cid:19) + d i +1 + O ( ε i +1 ) . In case b) we have µ ( B ( x, r )) ≤ Ξ (2 r ) d i +1 − ε i +1 and hence log µ ( B ( x, r ))log r ≥ log (cid:0) Ξ · (2 r ) d i +1 − ε i +1 (cid:1) log r . We again need to distinguish two cases: If d i +1 > d i , then(37) log µ ( B ( x, r ))log r ≥ log µ ( B ( x, R i ))log R i ≥ d i + O ( ε i , ε i +1 ) , while in the case d i ≥ d i +1 we have(38) log µ ( B ( x, r ))log r ≥ log µ ( B ( x, R i +1 ))log R i +1 ≥ d i +1 + O (cid:18) n i n i +1 (cid:19) + O ( ε i , ε i +1 ) . The estimations (36), (37), and (38) prove that if the sequence { n i } i in-creases fast enough then for every x ∈ A we have d µ ( x ) ≥ lim inf i →∞ d i and d µ ( x ) ≥ lim sup i →∞ d i . Hence, applying the Frostman lemma, we obtaindim H A ≥ lim inf i →∞ d i and dim P A ≥ lim sup i →∞ d i , and hence the assertion of Theorem 3 follows. (cid:3) References
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