The Lyapunov spectrum of some parabolic systems
aa r X i v : . [ m a t h . D S ] S e p THE LYAPUNOV SPECTRUM OF SOME PARABOLICSYSTEMS
KATRIN GELFERT AND MICHA L RAMS
Abstract.
We study the Hausdorff dimension spectrum for Lyapunovexponents for a class of interval maps which includes several non-hyper-bolic situations. We also analyze the level sets of points with given lowerand upper Lyapunov exponents and, in particular, with zero lower Lya-punov exponent. We prove that the level set of points with zero exponenthas full Hausdorff dimension, but carries no topological entropy. Introduction
Our goal here is to present results on the Lyapunov spectrum of intervalmaps with parabolic periodic points. We are going to work in the followingsetting.Let f : I → I be a map on some interval I ⊂ R for which there is apartition I = I ∪ . . . ∪ I ℓ into sub-intervals such that f | I i is monotone andcontinuously differentiable for every i . Let Λ ⊂ I be a compact f -invariantset such that f | Λ is topologically conjugate to a topologically mixing subshiftof finite type. Assume that f | Λ satisfies the tempered distortion property(see Definition 2.1 for the definition). Let (Λ m ) m be an increasing family ofcompact f -invariant sets having the property that f | Λ m has bounded dis-tortion and is uniformly expanding and topologically conjugate to subshiftsof finite type, and that Λ m converges to Λ in the Hausdorff topology.Our goal is to study the spectrum of Lyapunov exponents of such systems.Given x ∈ Λ we denote by χ ( x ) and χ ( x ) the lower and upper Lyapunovexponent at x , respectively, χ ( x ) def = lim inf n →∞ n log | ( f n ) ′ ( x ) | χ ( x ) def = lim sup n →∞ n log | ( f n ) ′ ( x ) | , and if both values coincide then we call the common value the Lyapunovexponent at x and denote it by χ ( x ). For given numbers 0 ≤ α ≤ β we Mathematics Subject Classification.
Primary: 37E05, 37D25, 37C45, 28D99 .
Key words and phrases.
Lyapunov exponents, multifractal spectra, Hausdorff dimen-sion, nonuniformly hyperbolic systems.This research of K. G. was supported by the grant EU FP6 ToK SPADE2 and by theDeutsche Forschungsgemeinschaft. The research of M. R. was supported by grants EUFP6 ToK SPADE2, EU FP6 RTN CODY and MNiSW grant ’Chaos, fraktale i dynamikakonforemna’. consider the following level sets L ( α, β ) def = { x ∈ Λ : χ ( x ) = α, χ ( x ) = β } . If α < β then L ( α, β ) is contained in the set of so-called irregular points L irr def = (cid:8) x ∈ Λ : χ ( x ) < χ ( x ) (cid:9) . It follows from the Birkhoff ergodic theorem that then we have µ ( L irr ) = 0for any f -invariant probability measure µ supported on Λ. We denote by L ( α ) def = L ( α, α ) the set of regular points with exponent α . Similarly, given0 ≤ α ≤ β , β > b L ( α, β ) def = { x ∈ Λ : 0 < χ ( x ) ≤ β, χ ( x ) ≥ α } . Recall that the continuous function log | f ′ | : Λ → R is said to be cohomolo-gous to a constant if there exist a continuous function ψ : Λ → R and c ∈ R such that log | f ′ | = ψ − ψ ◦ f + c on Λ, which immediately implies that L ( c ) = Λ. From the following considerations we will exclude this trivialcase.We want to determine the complexity of these sets in terms of their Haus-dorff dimension dim H . The multifractal analysis of dynamical systems, in-cluding level sets of more general local quantities than the Lyapunov ex-ponents, are so far well understood only in the uniformly hyperbolic case(see [13] for main results and further references). Nevertheless, we can men-tion several results beyond the hyperbolic setting. Nakaishi [10] studiedManneville-Pomeau-like maps and derived the Hausdorff dimension of thelevel sets L ( α ) for Lyapunov exponents α in the interior of the spectrum.Similar results for a different map was obtained by Kesseb¨ohmer and Strat-mann [9].In many approaches to a multifractal analysis of such level sets one char-acterizes their dimension (or their entropy) in terms of a conditional varia-tional principle of dimension (or entropies) of measures. We prefer insteada description which involves the Legendre-Fenchel transform of the pressurefunction. To start with our general scheme, it would be desirable to obtainin the above setting a formula for the dimension spectrum of the Birkhoffaverages of a general continuous (or H¨older continuous) potential ϕ : Λ → R ,that is, to prove for suitable values β for example thatdim H (cid:26) x : lim n →∞ n (cid:16) ϕ ( x ) + . . . + ϕ ( f n − ( x )) (cid:17) = β (cid:27) = 1 β sup d ∈ R ( dβ − P ( dϕ )) , generalizing nowadays classical results (see, e.g. [1], where, however, theonly considered values are in the interior of the interval of all the possibleaverages β ). In the present paper we will investigate the particular case ofthe potential ϕ = − log | f ′ | . Let us denote(1) F ( α ) def = 1 α inf d ∈ R (cid:0) P ( − d log | f ′ | ) + αd (cid:1) and let(2) F (0) def = lim α → F ( α ) = d , where d def = inf { d : P ( − d log | f ′ | ) = 0 } (see Section 2.2 for fundamental properties of F ).The following is our first main result. Theorem 1.
Under the conditions above, for all ≤ α ≤ β , β > , forwhich b L ( α, β ) , L ( α, β ) are nonempty we have dim H b L ( α, β ) = max α ≤ q ≤ β F ( q ) and dim H L ( α, β ) = min α ≤ q ≤ β F ( q ) . The above formulas extend what is known in the hyperbolic setting inseveral aspects. First of all, it applies to several non-hyperbolic situations.Second, we are able to cover the boundary points of the Lyapunov spectrum(see [15, 14] for related results in the case of the topological entropy of levelsets). Finally, we give a description of the dimension of level sets containingirregular points with zero lower Lyapunov exponent. It generalizes resultsof Barreira and Schmeling [2].Of particular interest is the set L (0). If f | Λ satisfies the specificationproperty, then the entropy spectrum of Birkhoff averages of general con-tinuous potentials have been studied in [15] using a different approach, seealso [14]. For such a system, the vanishing of the entropy h top ( f | L (0)) asstated below follows in fact from [15, Theorem 3.5] and the Ruelle inequal-ity. (Here we note that L (0) need not to be compact, and we are using thenotion of topological entropy on non-compact sets introduced by Bowen,see Section 6).On the other hand, in terms of Hausdorff dimension L (0) isa rather large set. Theorem 2. If L (0) is nonempty then we have dim H L (0) = dim H Λ ≥ F (0) and h top ( f | L (0)) = 0 . We now sketch the exposition of our paper. In Section 2 we review sev-eral concepts and results from ergodic theory. In Section 3 we analyze themain properties of the hyperbolic sub-systems which we are going to con-sider. Upper bounds for the dimension are studied in Section 4. Section 5is devoted to the analysis of lower dimension bounds: for the set of regularpoints with an exponent from the interior of the spectrum such bounds sim-ply follow from the maximal lower bound for the corresponding hyperbolicsub-systems. In order to handle exponents at the boundary of the spectrumas well as a set of irregular points, we introduce the concept of a w-measureas the main tool of our analysis. The proofs of Theorems 1 and 2 are givenat the end of Section 5 and in Section 6.
KATRIN GELFERT AND MICHA L RAMS Preliminaries
Examples.
Before we collect some examples, let us introduce somenotation. Consider the topological Markov chain σ : Σ → Σ defined by σ ( i i . . . ) = ( i i . . . ) on the setΣ def = { , . . . , p } N The inverse branches of σ will be denoted by σ i . We denote Σ n = { , . . . , p } n and Σ ∗ = S ∞ n =0 Σ n , where we use the convention Σ = { ∅ } .We will assume that f | Λ is topologically conjugate to a topologicallymixing subshift of finite type (Σ A , σ ) ⊂ (Σ , σ ).Let I , . . . , I p be a family of compact subintervals of I with pairwisedisjoint interiors and assume that g i ( I ) ⊂ I i , where g i is the inverse branchof f , conjugate to σ i . For each ( i . . . i n ) ∈ Σ n we define∆ i ...i n def = g i ...i n − ( I i n )and ∆ ∅ = I . Given x ∈ Λ, we denote a cylinder ∆ i ...i n containing x alsoby ∆ n ( x ).Our standing assumption is the tempered distortion property of f : Definition.
The map f has tempered distortion on Λ if there exists a pos-itive sequence ( ρ n ) n decreasing to 0 such that for every n we have(3) sup ( i ...i n ) sup x,y ∈ ∆ i ...in | ( f n ) ′ ( x ) || ( f n ) ′ ( y ) | ≤ e nρ n . We say that f is uniformly expanding or uniformly hyperbolic on an f -invariant compact set K ⊂ Λ if there exists c > λ > | ( f n ) ′ | ≥ cλ n everywhere on K . There are two main classes of (nonuniformlyhyperbolic) examples we can work with. The first class is closely related toparabolic Cantor sets, introduced in [16]. Example 1 (parabolic IFS) . Assume that | f ′ | > p i where | f ′ ( p i ) | = 1. Assume also that f is C s for some positive s . We construct the subsystems Λ m by removing somesmall cylinder neighborhoods of parabolic points and all their pre-images.Those subsystems are hyperbolic and have bounded distortion.This class of examples contains for example the celebrated Manneville-Pomeau maps [11]: f : [0 , → [0 ,
1] : x x (1 + x s ) mod 1, s > Remark 1.
Strictly speaking, the Manneville-Pomeau map is not conju-gated to a subshift of finite type (some cylinders are only essentially disjoint,thus there exists a countable family of points belonging to two different cylin-ders of the same level). We will allow this situation, our proofs work in thiscase as well without major changes.The second class is related to the one introduced in [6].
Example 2 (expansive Markov systems) . Consider less restrictive assump-tions about f , demanding only thatlim n →∞ max i ...i n | ∆ i ...i n | = 0 . Assume also that f is piecewise C . The subsystems Λ m are constructed likein the previous case. Their hyperbolicity and bounded distortion propertyfollows from the Ma˜n´e hyperbolicity theorem, see [3] for the reference. Remark 2.
For both the above-mentioned classes of examples we haveequality in the assertion of Theorem 2.2.2.
Topological pressure.
Let ϕ be a continuous function on Λ. The topological pressure of ϕ (with respect to f | Λ) is defined by(4) P ( ϕ ) def = lim n →∞ n log X ( i ...i n ) exp max x ∈ ∆ i ...in S n ϕ ( x ) , where here and in the sequel the sum is taken over the cylinders with non-empty intersection with the set Λ. The existence of the limit follows easilyfrom the fact that the sum constitutes a sub-multiplicative sequence. More-over, the value P ( ϕ ) does not depend on the particular Markov partitionthat we use in its definition.Denote by M (Λ) the family of f -invariant Borel probability measures onΛ. We simply write M = M (Λ) if there is no confusion about the system.By the variational principle we have(5) P ( ϕ ) = max µ ∈ M (cid:18) h µ ( f ) + Z Λ ϕdµ (cid:19) , where h µ ( f ) denotes the entropy of f with respect to µ (see [17]). A measure µ ∈ M is called equilibrium state for the potential ϕ if P ( ϕ ) = h µ ( f ) + Z Λ ϕ dµ. Given d ∈ R , we define the function ϕ d : Λ Q → R by(6) ϕ d ( x ) def = − d log | f ′ ( x ) | . The tempered distortion property (3) ensures in particular that in the defi-nition of P ( ϕ d ) in (4) one can replace the maximum by the minimum or, infact, by any intermediate value. Proposition 1.
The function d P ( ϕ d ) is a continuous, convex, and non-increasing function of R . P ( ϕ d ) is negative for large d if and only if thereexist no f -invariant probability measures with zero Lyapunov exponent.Proof. The claimed properties follow immediately from general facts aboutthe pressure together with the variational principle (5). (cid:3)
Lemma 1.
We have d ≤ dim H Λ . KATRIN GELFERT AND MICHA L RAMS
Proof.
It follows immediately from the generator condition that every er-godic f -invariant measure ν has non-negative Lyapunov exponent. Supposenow that P ( − t log | f ′ | ) > t ≥
0. Then, by the variational princi-ple, there exists an ergodic f -invariant measure ν such that h ν ( f ) − tχ ( ν ) > h ν ( f ) >
0. It follows then from [7] that t < h ν ( f ) χ ( ν ) = dim H ν ≤ dim H Λ . (cid:3) We now give a geometric description of F defined in (1), (2) for positive α . In general, F is always a concave function with range {−∞} ∪ [0 , ∞ ).Let us write P ( d ) def = P ( − d log | f ′ | ). Note that, by continuity and convexity,the pressure function d P ( d ) may fail to be differentiable on an at mostcountable set. We sketch below the particular case that we may have atmost one point of non-differentiability. If P ′ ( d ) = − α then F ( α ) = P ( d ) + αdα . If P ( d ) ≥ d , that is, the system is parabolic, then we have F (0) = inf { s ≥ P ( s ) = 0 } . In this situation we have the following two possible cases.Case I: The pressure function is differentiable at F (0). Then F ( α ) is strictlydecreasing for positive α .Case II: The pressure function is not differentiable at F (0). Then F ( α ) = F (0) for every α ≤ − lim s → F (0) − P ′ ( s ), and F is strictly decreasing forgreater α .2.3. Conformal measures.
The
Ruelle-Perron-Frobenius transfer opera-tor L ϕ : C (Λ) → C (Λ) defined on the space C (Λ) of continuous functions ψ : Λ → R is given by L ϕ ψ ( x ) def = X f ( y ) = x,y ∈ Λ e ϕ ( y ) ψ ( y )if x ∈ Λ. Denote by λ ϕ the spectral radius of L ϕ . Let ν be an eigenmeasureof the dual operator L ∗ ϕ with eigenvalue λ ϕ . Note that ν is a probabilitymeasure but not necessarily f -invariant. However, the dynamical propertiesof ν with respect to f | Λ are captured through its Jacobian. The
Jacobian of ν with respect to f | Λ is the (essentially) unique function J ν f determinedthrough(7) ν ( f ( A )) = Z A J ν f dν. for every Borel subset A of Λ such that f | A in injective, and is given by J ν f = λ ϕ e − ϕ (see [18]). Moreover, by [19, Theorem 2.1] we have log λ ϕ = PSfrag replacements ∼ − dα + ∼ − dα − α − α + P ( d ) dim H Λdim H Λ d ∼ − dα α −∞ F ( α ) α Figure 1.
Pressure and Lyapunov spectrum for uniformlyhyperbolic systemPSfrag replacements ∼ − dα + ∼ − dα − α − α + P ( d ) dim H Λdim H Λ d ∼ − dα α −∞ F ( α ) α Figure 2.
Pressure and Lyapunov spectrum for parabolicsystem, Case IPSfrag replacements ∼ − dα + ∼ − dα − α − α + P ( d ) dim H Λ dim H Λ d ∼ − dα α −∞ F ( α ) α Figure 3.
Pressure and Lyapunov spectrum for parabolicsystem, Case II P ( ϕ ). A measure satisfying (7) is called e P ( ϕ ) − ϕ -conformal measure . Sucha measure always exists if f | Λ is expansive and open ([4, Theorem 3.12]).3.
Hyperbolic sub-systems
For shortness, we will write P m = P f | Λ m . Given m , the function d P m ( ϕ d ) is analytic and strictly decreasing. Forfixed d ∈ R , the sequence P m ( ϕ d ) is non-decreasing. Proposition 2.
Given d ∈ R , we have P ( ϕ d ) = lim m →∞ P m ( ϕ d ) . KATRIN GELFERT AND MICHA L RAMS
Proof.
Assume that this is not the case for some d ∈ R . Clearly, P m ( ϕ d ) forman increasing sequence. Notice that P ∗ ( d ) def = lim m →∞ P m ( ϕ d ) ≤ P ( ϕ d ). Let δ def = P ( ϕ d ) − P ∗ ( d ). Let m ≥ P m ( ϕ d ) ≥ P ∗ ( d ) − δ/ m ≥ m .There exists a sequence of exp ( P m ( ϕ d ) − ϕ d )-conformal measures (withrespect to the sub-system f | Λ m ) which we denote by ν md . Each such measuresatisfies 1 = ν md ( f n (∆ n ( x ))) = Z ∆ n ( x ) e nP m ( ϕ d ) | ( f n ) ′ ( y ) | d dν md ( y )for every n ≥ x ∈ Λ m . Hence, from the tempered distortionproperty (3) we can conclude that e − nρ n ≤ ν md (∆ n ( x ))exp ( − nP m ( ϕ d )) | ( f n ) ′ ( x ) | − d ≤ e nρ n . Notice that this inequality holds only for cylinders ∆ n ( x ) which intersectsΛ m . However, if ∆ n ( x ) intersects Λ, then it intersects Λ m for every m sufficiently big.Likewise for f | Λ we obtain for the exp ( P ( ϕ d ) − ϕ d )-conformal measure e − nρ n ≤ ν d (∆ n ( x ))exp ( − nP ( ϕ d )) | ( f n ) ′ ( x ) | − d ≤ e nρ n for every cylinder ∆ n ( x ) intersecting Λ. Hence, we obtain for every n ≥ i ...i n ( x ) which intersects Λ m . ν d (∆ i ...i n ) ≤ ν md (∆ i ...i n ) e − n ( P ( ϕ d ) − P m ( ϕ d )) e nρ n ≤ ν md (∆ i ...i n ) e − nδ e nρ n for every m ≥
1. Take a subsequence ( ν m k d ) k converging to some probabilitymeasure ν ∗ d in the weak ∗ topology. Then we obtain ν d (∆ i ...i n ) ≤ ν ∗ d (∆ i ...i n ) e n (2 ρ n − δ ) < ν ∗ d (∆ i ...i n )for every ( i . . . i n ) ∈ Σ Q m ,n . This contradicts the fact that both measuresare probability measures. (cid:3) We introduce some further notation. Let α − m def = inf { α ≥ χ ( x ) = α for some x ∈ Λ m } ,α + m def = sup { α ≥ χ ( x ) = α for some x ∈ Λ m } . Similarly, let α − def = inf { α ≥ χ ( x ) = α for some x ∈ Λ } ,α + def = sup { α ≥ χ ( x ) = α for some x ∈ Λ } . Those are easy to calculate using the pressure, since we have α − = lim d →∞ − d P ( ϕ d ) ,α + = lim d →−∞ − d P ( ϕ d ) . Lemma 2.
We have lim m →∞ α − m = inf m ≥ α − m = α − , lim m →∞ α + m = sup m ≥ α + m = α + . Proof.
We have for d < P m ( ϕ d ) + dα + m ≤ P m (0) ≤ P (0) , hence, by Proposition 2, we obtain α + = lim d →−∞ − d P ( ϕ d ) ≤ sup m ≥ α + m . Similarly, for d > m big enough we have P m ( ϕ d ) ≤ P m (0) − dα − m ≤ P (0) − dα − m , and thus α − = lim d →∞ − d P ( ϕ d ) ≥ inf m ≥ α − m . The opposite inequalities follow from the definition of α ± and α ± m and fromΛ m ⊂ Λ. (cid:3) Given α > m ≥ F m ( α ) def = 1 α inf d ∈ R ( P m ( ϕ d ) + αd ) . Lemma 3.
For every α ∈ ( α − , α + ) we have (8) sup m ≥ F m ( α ) = lim m →∞ F m ( α ) = F ( α ) . Proof.
First notice that we can rewrite F ( α ) = 1 α inf d ∈ R ( P ( ϕ d ) + αd )= sup { d : dα ≤ P ( ϕ s ) + sα for every s } . The analogous relation holds for F m with P replaced by P m . Let us assumethat there exists ε > F m ( α ) < F ( α ) − ε for every m ≥
1. Thiswould imply that for every m the set J m def = { s : P m ( ϕ s ) + sα ≤ ( F ( α ) − ǫ ) α } is non-empty, closed, and bounded. Moreover, as P m +1 ≥ P m , we have J m +1 ⊂ J m . Hence, T m ≥ J m is non-empty. For s ∈ T m ≥ J m we concludethat P m ( ϕ s ) + sα ≤ ( F ( α ) − ε ) α for every m ≥
1. Together with Proposition 2 we hence would obtain P ( ϕ s ) + sα ≤ ( F ( α ) − ε ) α which is a contradiction. We mention a second way of proving (8) which is based on the convexconjugate functions. Let T m ( α ) def = sup d ∈ R ( αd − P m ( ϕ − d ))denote the convex conjugate of d P m ( ϕ − d ). Then ( P m , T m ) form aLegendre-Fenchel pair. Wijsman [12] has shown that for given Legendre-Fenchel pairs ( P m , T m ) and ( P, T ), the functions P m converge infimally to P if and only if T m converges infimally to T (we refer to [12] for the definition ofinfimal convergence). In general, this kind of convergence does not coincidewith the pointwise convergence. However, by monotonicity and continuityof the pressure function we obtain that P m converges infimally if and onlyif it converges pointwise. The application of Proposition 2 implies (8). (cid:3) For the remainder of this section let K ⊂ Λ be some f -invariant compactset such that f | K is uniformly expanding. We have the following result byJenkinson [8]. Lemma 4 ([8]) . For any α ∈ (inf ν ∈ M ( K ) χ ( ν ) , sup ν ∈ M ( K ) χ ( ν )) there existsa number q = q ( α ) and some equilibrium state ν = ν ( α ) for the potential q log | f ′ | (with respect to f | K ) such that Z K log | f ′ | dν = α. We finally collect results on the dimension of level sets for hyperbolicsystems.
Proposition 3.
For every α ∈ (inf ν ∈ M ( K ) χ ( ν ) , sup ν ∈ M ( K ) χ ( ν )) we have (9) dim H ( K ∩ L ( α )) = 1 α inf d ∈ R (cid:0) P f | K ( ϕ d ) + dα (cid:1) . Proof.
From [1, Theorem 6] it follows that for arbitrary d ∈ R dim H ( K ∩ L ( α )) = max (cid:26) h ν ( f ) R K log | f ′ | dν : ν ∈ M ( K ) , Z K log | f ′ | dν = α (cid:27) = 1 α max (cid:26) h ν ( f ) + d Z K log | f ′ | dν : ν ∈ M ( K ) , χ ( ν ) = α (cid:27) − d ≤ α max (cid:26) h ν ( f ) + d Z K log | f ′ | dν : ν ∈ M ( K ) (cid:27) − d = 1 α (cid:0) P f | K ( d log | f ′ | ) − dα (cid:1) , where we applied the variational principle for the topological pressure. Sowe obtain dim H ( K ∩ L ( α )) ≤ α inf d ∈ R (cid:0) P f | K ( ϕ d ) + dα (cid:1) . Lemma 4 implies thatmax { h ν ( f ) : ν ∈ M ( K ) , χ ( ν ) = α } ≥ P f | K ( − q log | f ′ | ) + qα ≥ inf d ∈ R (cid:0) P f | K ( ϕ d ) + dα (cid:1) . This finishes the proof. (cid:3)
Given any H¨older continuous potential ψ : K → R , there exists a uniqueergodic equilibrium state µ ∈ M ( K ) which moreover has the Gibbs property,that is, for which there exists a constant c > x ∈ K andevery n ≥ c − ≤ µ (∆ n ( x ) ∩ K )exp (cid:0) − nP f | K ( ψ ) + S n ψ ( x ) (cid:1) ≤ c . We refer for example to [13] for more details and references of the aboveresults. 4.
Upper bound for the dimension
Proposition 4.
We have for every ≤ α ≤ β , β > H b L ( α, β ) ≤ max α ≤ q ≤ β F ( q ) and dim H L ( α, β ) ≤ min α ≤ q ≤ β F ( q ) . Proof.
Note first that L ( α, β ) ⊂ b L ( q, q ) for any q ∈ [ α, β ] \ { } . Hence, thesecond assertion follows from the first one.We now prove the first assertion. For a point x ∈ b L ( α, β ) there exists apositive number q = q ( x ) ∈ [ α, β ] and a sequence ( n k ) k for which we have(11) lim k →∞ n k log | ( f n k ) ′ ( x ) | = q Let δ ∈ (0 , q ). There exists N = N ( x ) ≥ e n k ( q − δ ) ≤ | ( f n k ) ′ ( x ) | ≤ e n k ( q + δ ) for every n k ≥ N . By the tempered distortion property (3) we obtain(13) | Λ | | ( f n k ) ′ ( x ) | − e − n k ρ nk ≤ | ∆ n k ( x ) | ≤ | Λ | | ( f n k ) ′ ( x ) | − e n k ρ nk . Using again the tempered distortion property (3) we can conclude that theexp ( P ( ϕ d ) − ϕ d )-conformal measure satisfies(14) e − nP ( ϕ d ) | ( f n ) ′ ( x ) | − d e − nρ n ≤ ν d (∆ n ( x )) ≤ e − nP ( ϕ d ) | ( f n ) ′ ( x ) | − d e nρ n . We obtain(15) P ( ϕ d ) + lim k →∞ n k log ν d (∆ n k ( x )) = − d q, and in particular the limit on the left hand side exists. Hence, possibly afterincreasing N , we have e − n k ( P ( ϕ d )+ dq + δ ) ≤ ν d (∆ n k ( x ))for every n k ≥ N . With (13) and (14) we can conclude that ν d (∆ n k ( x )) ≥ e − n k P ( ϕ d ) | ∆ n k ( x ) | d | Λ | − d e − n k ρ nk (1+ | d | ) . Case 1) Let us first assume that P ( φ d ) ≥
0. Using (12) and (13) we canestimate e − n k P ( ϕ d ) ≥ (cid:0) | ∆ n k ( x ) | | Λ | − e − n k ρ nk (cid:1) P ( ϕ d ) / ( q − δ ) . Thus we obtain ν d (∆ n k ( x )) ≥ (cid:0) | ∆ n k ( x ) | | Λ | − (cid:1) d + P ( ϕd ) q − δ e − n k ρ nk “ P ( ϕd ) q − δ +(1+ | d | ) ” . There exists ε = ε ( δ ) > N again, wehave | Λ | − d − P ( ϕd ) q − δ e − n k ρ nk “ P ( ϕd ) q − δ +(1+ | d | ) ” ≥ | ∆ n k ( x ) | ε for every n k ≥ N . Note that ∆ n k ( x ) ⊂ B ( x, | ∆ n k ( x ) | ). Hence, we obtainthe following upper bound for the lower pointwise dimension at x (16) d ν d ( x ) ≤ P ( ϕ d ) q − δ + d + ε. Case 2) Let us now assume that P ( ϕ d ) <
0. Using (12) and (13) we canestimate e − n k P ( ϕ d ) ≥ (cid:0) | ∆ n k ( x ) || Λ | − (cid:1) P ( ϕd ) q + δ e n k ρ nk P ( ϕd ) q + δ . Thus we obtain ν d (∆ n k ( x )) ≥ | ∆ n k ( x ) | d + P ( ϕd ) q + δ + ε for every n k ≥ N , possibly after increasing N , and hence in this case(17) d ν d ( x ) ≤ P ( ϕ d ) q + δ + d + ε. In both cases, continuity of d P ( ϕ d ) implies that for any given suffi-ciently small interval ( q ′ , q ′′ ) ⊂ (max { , α − δ } , β + δ ) there exist d ∈ R suchthat 1 q ′′ P ( ϕ d ) + d ≤ F ( q ′′ ) + ε. We can then choose a countable family of intervals ( q ′ i , q ′′ i ), covering (max { , α − δ } , β + δ ), and consider the corresponding sequence ( d i ) i . Define ν def = ∞ X i =1 − i ν d i We have d ν ( x ) ≤ sup i ≥ d ν di ( x ) ≤ max α − δ ≤ q ≤ β + δ F ( q ) + 2 ε, where the second inequality follows from (16), (17). This implies thatdim H b L ( α, β ) ≤ max α − δ ≤ q ≤ β + δ F ( q ) + 2 ε. Since δ and ε can be chosen arbitrarily small, this finishes the proof. (cid:3) Given α > b L ( α ) def = (cid:8) x ∈ Λ : χ ( x ) = 0 , χ ( x ) ≥ α (cid:9) . The following proposition is proved in a similar way to Proposition 4.
Proposition 5.
We have for every α > H b L ( α ) ≤ F ( α ) . Lower bound for the dimension
The interior of the spectrum – regular points.Proposition 6.
For α > α − we have dim H L ( α ) ≥ F ( α ) . Proof.
Denote H m ( α ) def = L ( α ) ∩ Λ m . For each exponent α > α − there exists m ≥ α > α − m and hence α > α − m ′ for every m ′ ≥ m . ByProposition 3 we have F f | Λ m ′ ( α ) = dim H H m ′ ( α ), and we can conclude thatdim H L ( α ) ≥ sup m ≥ dim H H m ( α ) = sup m ≥ F f | Λ m ( α ) . The application of Lemma 3 finishes the proof. (cid:3)
Construction of w-measures and their properties.
Recall the no-tation for hyperbolic sub-systems introduced in Section 3. Given a nonde-creasing sequence of positive integers ( n i ) i , let ( µ i ) i be a sequence of certainequilibrium states for potentials φ i with respect to f | Λ n i . We denote h i def = h µ i ( f ) , χ i def = χ ( µ i ) , d i def = h i χ i = dim H µ i . (Note that the last equality uses a result in [7].) We note that the same con-struction can be performed for an arbitrary, not necessarily non-decreasing,sequence ( n i ) i . But this assumption simplifies the exposition. We will inthe following assume that(19) P f | Λ ni ( φ i ) = 0(note that otherwise we can replace φ i by φ i − P f | Λ ni ( φ i ) without changingthe equilibrium state µ i ).We now describe the construction of a measure µ , satisfying certain specialproperties. Let ( m i ) i be a fast increasing sequence of positive integers. Wewill specify the specific growth speed in the course of this section. Wedemand that(20) ρ m i χ i → , where ( ρ m ) m is a positive sequence decreasing to 0 as in (3). We define aprobability measure µ on the algebra generated by the cylinders ∆ i ...i m . Asthe beginning of the construction, for cylinders of level m we define µ (∆ ω m ) def = µ (∆ ω m ) . Given a cylinder of level m i of positive measure µ , we sub-distribute themeasure on its sub-cylinders of level m i +1 which intersect Λ m i +1 in the fol-lowing way. Let µ (∆ ω mi τ mi +1 − mi ) def = c i +1 ( ω m i ) µ (∆ ω mi ) µ i +1 (∆ τ mi +1 − mi )where c i +1 ( ω m i ) = X τ mi +1 − mi : ω mi τ mi +1 − mi ∈ Σ ni +1 µ i +1 (∆ τ mi +1 − mi ) − is the normalizing constant. For every m i < m < m i +1 let µ (∆ π m ) def = X τ mi +1 − m µ (∆ π m τ mi +1 − m ) . We extend the measure µ arbitrarily to the Borel σ -algebra of Λ. Wecall the probability measure µ a w-measure with respect to the sequence( f | Λ n i , φ i , µ i ) i . We will in the following analyze the precise way in whichthe Lyapunov exponents and the local entropies of µ are determined by theasymptotic fluctuations of the Lyapunov exponents and the local entropiesof the equilibrium states µ i .As each µ i +1 is an equilibrium state for a uniformly hyperbolic system,it has the Gibbs property (10) with some constant D i +1 . Hence we canconclude that D − i +1 ≤ µ i +1 (∆ ω mi ) X τ mi +1 − mi : ω mi τ mi +1 − mi ∈ Σ ni +1 µ i +1 (∆ ω mi τ mi +1 − mi ) c i +1 ( ω m i ) ≤ D i +1 . Now observe that µ i +1 (∆ ω mi ) − [ · · · ] = 1 and hence D − i +1 ≤ c i +1 ( ω m i ) ≤ D i +1 . Notice further that the Gibbs property (10) implies that(21) D − i +1 ≤ µ i +1 (∆ ω mi τ m − mi ) µ i +1 (∆ ω mi ) µ i +1 (∆ τ m − mi ) ≤ D i +1 . Hence we obtain for the constructed measure µ (22) D − i +1 µ (∆ ω mi ) µ i +1 (∆ ω mi ) ≤ µ (∆ ω mi τ mi +1 − mi ) µ i +1 (∆ ω mi τ mi +1 − mi ) ≤ D i +1 µ (∆ ω mi ) µ i +1 (∆ ω mi ) . Consider now the measure of a cylinder at level m for any m i < m < m i +1 .Notice that µ i +1 (∆ π m ) = X τ mi +1 − m µ i +1 (∆ π m τ mi +1 − m ) . Hence, (22) implies that for any m i < m < m i +1 (23) D − i +1 µ (∆ ω mi ) µ i +1 (∆ ω mi ) ≤ µ (∆ ω mi τ m − mi ) µ i +1 (∆ ω mi τ m − mi ) ≤ D i +1 µ (∆ ω mi ) µ i +1 (∆ ω mi ) . Now (23) and (21) imply and(24) D − i +1 ≤ µ (∆ ω mi τ m − mi ) µ (∆ ω mi ) µ i +1 (∆ τ m − mi ) ≤ D i +1 for any m i < m ≤ m i +1 and any sequence ω m i τ m − m i such that ∆ ω mi τ m − mi intersects Λ i +1 .Denote L m ( x ) def = 1 m log | ( f m ) ′ ( x ) | and H m ( x ) def = − m log µ (∆ m ( x )) . Note that for m i < m ≤ m i +1 we have(25) L m ( x ) = m − m i m L m − m i ( f m i ( x )) + m i m L m i ( x ) . Further, by (24) for m i < m ≤ m i +1 we can estimate(26) (cid:12)(cid:12)(cid:12)(cid:12) H m ( x ) + 1 m log µ i +1 (∆ m − m i ( f m i ( x ))) − m i m H m i ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ D i +1 m i . Proposition 7.
For any ε > , denote (27) A H ( ε, m i ) def = n x : (cid:12)(cid:12)(cid:12)(cid:12) H m ( x ) − m i m H m i ( x ) − m − m i m h i +1 (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε | h i +1 − h i | for every m i < m ≤ m i +1 o . Then for any δ > there exists M ( δ ) ≥ such that for m i > M ( δ ) µ ( A H ( ε, m i )) > − δ Proof.
Fix some ε > δ >
0. We will prove that the set A H ( ε, m i ) ∩ ∆ ω mi has measure greater than 1 − δ µ (∆ ω mi ) for all ω m i when m i is bigenough. Denote C ( ε, m i , ω i ) = f m i ( A H ( ε, m i ) ∩ ∆ ω mi ) . As this set is a union of cylinders of level at most m i +1 − m i , by (24) it isenough to prove that µ ( C ( ε, m i , ω i )) ≥ − D − i +1 δ uniformly in ω m i for m i big enough.By (26), we obtain(28) (cid:12)(cid:12)(cid:12)(cid:12) H m ( x ) − m i m H m i ( x ) − m − m i m h i +1 (cid:12)(cid:12)(cid:12)(cid:12) ≤ D i +1 m i + m − m i m (cid:12)(cid:12)(cid:12)(cid:12) h i +1 + 1 m − m i log µ i +1 (∆ n ( y )) (cid:12)(cid:12)(cid:12)(cid:12) for y = f m i ( x ). From the Gibbs property of the measure µ i +1 we obtainthat | log µ i +1 (∆ n ( y )) − S n φ i +1 ( y ) | ≤ log D i +1 for any n ≥
1. Thus, the right hand side of (28) is not greater than W def = 9 log D i +1 m i + 1 m | ( m − m i ) h i +1 + S m − m i φ i +1 ( y ) | . The first summand is arbitrarily small for big m i . To estimate the secondone we note that it is a consequence of the Birkhoff ergodic theorem andthe Egorov theorem that for the given numbers ε > δ > N = N ( ε, δ ) such that we have µ i +1 (cid:18)(cid:26) x : (cid:12)(cid:12)(cid:12)(cid:12) S n φ i +1 ( x ) − n Z φ i +1 dµ i +1 (cid:12)(cid:12)(cid:12)(cid:12) ≤ nε ∀ n ≥ N (cid:27)(cid:19) ≥ − δ. From (19) and from the fact that µ i +1 is an equilibrium state we concludethat h i +1 = − Z φ i +1 dµ i +1 . Hence, for m i sufficiently big, W is smaller than any constant with ar-bitrarily big probability. In particular, for m i > N ( ε | h i +1 − h i | , D − i +1 δ ) itis smaller than ε | h i +1 − h i | with probability bigger than 1 − D − i +1 δ and theassertion follows. (cid:3) Proposition 8.
For any ε > , denote (29) A L ( ε, m i ) def = n x : (cid:12)(cid:12)(cid:12)(cid:12) L m ( x ) − m i m L m i ( x ) − m − m i m χ i +1 (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε | χ i +1 − χ i | for every m i < m ≤ m i +1 o . Then for any δ > there exists M ( δ ) ≥ such that for m i > M ( δ ) µ ( A L ( ε, m i )) ≥ − δ. Proof.
We note that if y ∈ ∆ m ( x ) then | L m ( x ) − L m ( y ) | ≤ ρ m , where( ρ m ) m ≥ is the to 0 decreasing sequence from the tempered distortion prop-erty (3). Now we can apply (20) and repeat the same reasoning as in theproof of Proposition 7. (cid:3) For any sequence ( ε i ) i ≥ we can choose a summable sequence ( δ i ) i ≥ anda sequence ( m i ) i ≥ such that Propositions 7 and 8 hold for the constructedmeasure µ . In such a situation, by the Borel-Cantelli lemma for µ -almostevery x ∈ Λ both (27) and (29) are satisfied for all except finitely many i .This leads us to the following proposition. Proposition 9. If ( m i ) i in the construction above increases sufficiently fast,then for µ -almost every x ∈ Λ we have i) lim inf m →∞ H m ( x ) = lim inf i →∞ h i , lim sup m →∞ H m ( x ) = lim sup i →∞ h i , ii) lim inf m →∞ L m ( x ) = lim inf i →∞ χ i , lim sup m →∞ L m ( x ) = lim sup i →∞ χ i , and iii) lim inf m →∞ H m ( x ) L m ( x ) = lim inf i →∞ d i .Moreover, dim H µ ≥ lim inf i →∞ d i . Proof.
Choose some ε >
0. Denote by A j def = \ i ≥ j ( A H ( ε i , m i ) ∩ A L ( ε i , m i )) . the set of points for which (27) and (29) are satisfied for all i ≥ j for somesequence ( ε i ) i ≥ (which will be specified in the following). Note that { A j } is an increasing family of sets and that S j ≥ A j is of full measure µ .Clearly H m ( x ) ∈ [ H − , H + ] for some 0 < H − , H + < ∞ independent of x (because there are only finitely many cylinders) and L m ( x ) ∈ [ L − , L + ]for some 0 < L − , L + < ∞ independent of x (because log | f ′ | is uniformlybounded on Λ).Let j ≥ µ ( A j ) >
0. Given x ∈ A j , by (27) we have (cid:12)(cid:12) H m j + k +1 ( x ) − h j + k +1 (cid:12)(cid:12) ≤ (cid:18) m j + k m j + k +1 + ε j + k (cid:19) | h j + k +1 − h j + k | + m j + k m j + k +1 | H m j + k ( x ) − h j + k | , and hence for every ℓ ≥ j we obtain (cid:12)(cid:12)(cid:12)(cid:12) H m ℓ +1 ( x ) h ℓ +1 − (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:18) m ℓ m ℓ +1 + ε ℓ (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) h ℓ h ℓ +1 − (cid:12)(cid:12)(cid:12)(cid:12) + m ℓ m ℓ +1 (cid:12)(cid:12)(cid:12)(cid:12) H m ℓ ( x ) h ℓ − (cid:12)(cid:12)(cid:12)(cid:12) h ℓ h ℓ +1 . Thus, if ε i is sufficiently small and if m i grows fast enough then we obtainthat H m i ( x ) h i → i → ∞ uniformly in x ∈ A j .Let I = I ( j ) be sufficiently big such that for all x ∈ A j and all i > I wehave H m i ( x ) h i ∈ (1 − ε, ε ) . By (27), for all m i < m ≤ m i +1 we have then(30) (cid:12)(cid:12)(cid:12)(cid:12) H m ( x ) − m i m h i − m − m i m h i +1 (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε i | h i +1 − h i | + ε. Using (29) instead of (27), we can, possibly after changing ( ε i ) i and ( m i ) i ,prove in an analogous way that for all m i < m ≤ m i +1 we have(31) (cid:12)(cid:12)(cid:12)(cid:12) L m ( x ) − m i m χ i − m − m i m χ i +1 (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε i | χ i +1 − χ i | + ε. Notice that h ( m ) def = m i m h i + m − m i m h i +1 satisfies h i ≤ h ( m ) ≤ h i +1 , which implies claim i) of the assertion. Claim ii)follows from (31) in an analogous way together with χ ( m ) def = m i m χ i + m − m i m χ i +1 . satisfying χ i ≤ χ ( m ) ≤ χ i +1 . As h ( m ) χ ( m ) ≥ min { d i , d i +1 } for all m i < m ≤ m i +1 , claim iii) of the assertion follows from (30) and (31).We finally prove the lower bound on the Hausdorff dimension of µ . Forall x ∈ A j and for all i ≥ I we have H m i ( x ) ≥ (1 − ε ) h i and L m i ( x ) ≤ (1 + ε ) χ i . Let e µ j be the restriction of µ to A j . For all x ∈ A j we have e µ j (∆ m ( x )) ≤ µ (∆ m i ( x )) = e − m i H mi ( x ) ≤ e − m i (1 − ε ) h i and | ∆ m i ( x ) | ≥ e − m i ρ mi e − m i L mi ( x ) ≥ e − m i (1+2 ε ) χ i for m i big enough, where we use (20) to obtain the second inequality. Let r i def = e − m i (1+2 ε ) χ i . We obtain e µ j ( B ( x, r i )) ≤ e − m i (1 − ε ) h i , and hence the lower pointwise dimension of e µ j at x is bounded by d e µ j ( x ) ≥ (1 − ε ) lim inf i →∞ d i . Since ε was arbitrary, we obtain d e µ j ( x ) ≥ lim inf i →∞ d i for every x ∈ A j .Thus, we can conclude that dim H µ ≥ lim inf i →∞ d i . (cid:3) The boundary of the spectrum – regular and irregular points.
The considerations in the previous section prove the following theorem.
Theorem 3.
Let ( n i ) i be a nondecreasing sequence of positive integers and ( µ i ) i be a sequence of equilibrium states for f | Λ n i . Then dim H (cid:26) x : 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 . We now are able to bound the dimension of irregular points from below.We also obtain bounds on the dimension of the level sets at the boundaryof the spectrum. Proposition 10.
For α − ≤ α ≤ β ≤ α + we have dim H b L ( α, β ) ≥ max α ≤ q ≤ β F ( q ) and dim H L ( α, β ) ≥ min α ≤ q ≤ β F ( q ) . In particular, we have dim H L ( α ) ≥ F ( α ) . Proof.
We consider some sequence ( a n ) n ≥ n of numbers α − n < a n < α + n suchthat lim n →∞ a n = α (by our assumptions, α − n < α + n for n big enough).By Lemma 4, there exists a sequence ( q n ) n ≥ of numbers and a sequence( ν n ) n ≥ of equilibrium states of the potentials q n log | f ′ | such that P f | Λ n ( q n log | f ′ | ) = h ν n ( f ) + q n a n . If we apply [7, Theorem 1] to each of the hyperbolic sub-systems f | Λ n weobtain thatdim H ν n = 1 a n (cid:0) P f | Λ n ( q n log | f ′ | ) − q n a n (cid:1) ≥ F f | Λ n ( a n )Note that continuity of the function F and Lemma 3 together imply thatlim n →∞ F f | Λ n ( a n ) = F ( α ) for properly chosen ( a n ). The application ofTheorem 3 finishes the proof. (cid:3) Proof of Theorem 1.
The assertions follow from Proposition 10 and Propo-sition 4. (cid:3)
Proof of Theorem 2.
Recall the definition of b L ( α ) in (18). To prove the firstpart of assertion we use the fact thatΛ \ L (0) = b L (0 , α + ) ∪ [ n ∈ N b L (2 − n α + )If L (0) = ∅ , then q F ( q ) is a non-increasing function and hence Proposi-tions 4 and 5 imply that the Hausdorff dimension of Λ \ L (0) is not greaterthan F (0). At the same time, by Proposition 10 we have dim H L (0) ≥ F (0).This implies that dim H Λ = dim H L (0) ≥ F (0).The second part of the assertion of Theorem 2 we will prove in the fol-lowing section. (cid:3) Topological entropy
We first briefly recall one more concept from the thermodynamic for-malism (for a detailed account we refer to [13]). Given a set Z ⊂ Λ (notnecessarily compact nor f -invariant) and numbers ε > n ∈ N , we denoteby M ε ( Z, n ) the maximal cardinality of a set of points in Z which belongto a ( n, ε )-separated set in Λ. We define the upper capacitive topologicalentropy of f on Z by(32) Ch ( f | Z ) = lim ε → lim sup n →∞ n log M ε ( Z, n ) . Analogously, we define the lower capacitive topological entropy of f on Z ,denoted by Ch ( f | Z ), by replacing the limes superior in (32) with the limesinferior. Since f | Λ is expansive, it in fact suffices for sufficiently small ε to take in (32) only the limit in n . We denote by h ( f | Z ) the topologicalentropy of f on Z , but we refer the reader to [13] for its precise definition.The following properties hold:1. Ch ( f | Z ) ≤ Ch ( f | Z ) and Ch ( f | Z ) ≤ Ch ( f | Z ) whenever Z ⊂ Z ⊂ Λ,2. h ( f | Z ) ≤ Ch ( f | Z ) ≤ Ch ( f | Z ),3. for a countable union Z = S k ∈I Z i we have h ( f | Z ) = sup k ∈I h ( f | Z i ).Moreover, when Z ⊂ Λ is f -invariant and compact then we have coincidencewith the classical topological entropy with respect to f | Z , that is, h ( f | Z ) = Ch ( f | Z ) = Ch ( f | Z ) . Recall that ∆ n ( x ) denotes the cylinder ∆ i ...i n containing x . Lemma 5. L (0) = { x ∈ Λ : lim sup n →∞ n log | ∆ n ( x ) | = 0 } .Proof. Given x ∈ Λ, we have f n (∆ n ( x )) ⊃ I i for some i ∈ { , . . . , p } .By the mean value theorem there exists y = y ( n ) ∈ ∆ n ( x ) such that | ( f n ) ′ ( y ) | | ∆ n ( x ) | ≥ I i and hence | I | − ≤ | ∆ n ( x ) | − ≤ | I i | − | ( f n ) ′ ( x ) | | ( f n ) ′ ( y ) || ( f n ) ′ ( x ) | . By the tempered distortion property we obtainlim sup n →∞ n log sup y ∈ ∆ n ( x ) | ( f n ) ′ ( y ) || ( f n ) ′ ( x ) | = 0 . from here the statement follows. (cid:3) Proposition 11.
We have h ( f | L (0)) = 0 .Proof. We are going to show the existence of a decreasing sequence of sets L k , all containing L (0), such that Ch ( f | L k ) decreases to 0.Given ε > N ∈ N we define the set L ε,N = { x ∈ Λ : | ∆ n ( x ) | ≥ (1 + ε ) − n for every n ≥ N } . Notice that L ε,N ⊂ L ε,N ′ for N ≤ N ′ and that by Lemma 5 L (0) ⊂ \ ε> [ N ∈ N L ε,N . For every x ∈ L ε,N we have | ∆ n ( x ) | ≥ (1 + ε ) − n for every n ≥ N . Hence,the number of ( n, ε )-separated sets needed to cover the set L ε,N is at most I (1 + ε ) n . From the definition of Ch we obtain Ch ( f | L ε,N ) ≤ log(1 + ε ).Now it follows that h ( f | L (0)) ≤ h ( f | [ N ∈ N L ε,N ) = sup N ∈ N h ( f | L ε,N ) ≤ sup N ∈ N Ch ( f | L ε,N ) ≤ log(1+ ε ) . Since ε is arbitrary, we can conclude h ( f | L (0)) = 0. (cid:3) References
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