The Lyapunov spectrum of some parabolic systems
We study the Hausdorff dimension spectrum for Lyapunov exponents for a class of interval maps which includes several non-hyperbolic situations. We also analyze the level sets of points with given lower and upper Lyapunov exponents and, in particular, with zero lower Lyapunov exponent. We prove that the level set of points with zero exponent has full Hausdorff dimension, but carries no topological entropy.
Key words and phrases:Lyapunov exponents, multifractal spectra, Hausdorff dimension, nonuniformly hyperbolic systems
2000 Mathematics Subject Classification:Primary: 37E05, 37D25, 37C45, 28D99
Our goal here is to present results on the Lyapunov spectrum of interval maps with parabolic periodic points. We are going to work in the following setting.
Let be a map on some interval for which there is a partition into sub-intervals such that is monotone and continuously differentiable for every . Let be a compact -invariant set such that is topologically conjugate to a topologically mixing subshift of finite type. Assume that satisfies the tempered distortion property (see Definition 2.1 for the definition). Let be an increasing family of compact -invariant sets having the property that has bounded distortion and is uniformly expanding and topologically conjugate to subshifts of finite type, and that converges to in the Hausdorff topology.
Our goal is to study the spectrum of Lyapunov exponents of such systems. Given we denote by and the lower and upper Lyapunov exponent at , respectively,
and if both values coincide then we call the common value the Lyapunov exponent at and denote it by . For given numbers we consider the following level sets
If then is contained in the set of so-called irregular points
It follows from the Birkhoff ergodic theorem that then we have for any -invariant probability measure supported on . We denote by the set of regular points with exponent . Similarly, given , we will study
Recall that the continuous function is said to be cohomologous to a constant if there exist a continuous function and such that on , which immediately implies that . From the following considerations we will exclude this trivial case.
We want to determine the complexity of these sets in terms of their Hausdorff dimension . The multifractal analysis of dynamical systems, including level sets of more general local quantities than the Lyapunov exponents, are so far well understood only in the uniformly hyperbolic case (see  for main results and further references). Nevertheless, we can mention several results beyond the hyperbolic setting. Nakaishi  studied Manneville-Pomeau-like maps and derived the Hausdorff dimension of the level sets for Lyapunov exponents in the interior of the spectrum. Similar results for a different map was obtained by Kesseböhmer and Stratmann .
In many approaches to a multifractal analysis of such level sets one characterizes their dimension (or their entropy) in terms of a conditional variational principle of dimension (or entropies) of measures. We prefer instead a description which involves the Legendre-Fenchel transform of the pressure function. To start with our general scheme, it would be desirable to obtain in the above setting a formula for the dimension spectrum of the Birkhoff averages of a general continuous (or Hölder continuous) potential , that is, to prove for suitable values for example that
generalizing nowadays classical results (see, e.g. , where, however, the only considered values are in the interior of the interval of all the possible averages ). In the present paper we will investigate the particular case of the potential . Let us denote
(see Section 2.2 for fundamental properties of ).
The following is our first main result.
Under the conditions above, for all , , for which , are nonempty we have
The above formulas extend what is known in the hyperbolic setting in several 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 level sets). Finally, we give a description of the dimension of level sets containing irregular points with zero lower Lyapunov exponent. It generalizes results of Barreira and Schmeling .
Of particular interest is the set . If satisfies the specification property, then the entropy spectrum of Birkhoff averages of general continuous potentials have been studied in  using a different approach, see also . For such a system, the vanishing of the entropy as stated below follows in fact from [15, Theorem 3.5] and the Ruelle inequality. (Here we note that need not to be compact, and we are using the notion of topological entropy on non-compact sets introduced by Bowen, see Section 6).On the other hand, in terms of Hausdorff dimension is a rather large set.
If is nonempty then we have
We now sketch the exposition of our paper. In Section 2 we review several concepts and results from ergodic theory. In Section 3 we analyze the main properties of the hyperbolic sub-systems which we are going to consider. Upper bounds for the dimension are studied in Section 4. Section 5 is devoted to the analysis of lower dimension bounds: for the set of regular points with an exponent from the interior of the spectrum such bounds simply follow from the maximal lower bound for the corresponding hyperbolic sub-systems. In order to handle exponents at the boundary of the spectrum as well as a set of irregular points, we introduce the concept of a w-measure as the main tool of our analysis. The proofs of Theorems 1 and 2 are given at the end of Section 5 and in Section 6.
Before we collect some examples, let us introduce some notation. Consider the topological Markov chain defined by on the set
The inverse branches of will be denoted by . We denote and , where we use the convention .
We will assume that is topologically conjugate to a topologically mixing subshift of finite type .
Let , , be a family of compact subintervals of with pairwise disjoint interiors and assume that , where is the inverse branch of , conjugate to . For each we define
and . Given , we denote a cylinder containing also by .
Our standing assumption is the tempered distortion property of :
The map has tempered distortion on if there exists a positive sequence decreasing to such that for every we have
We say that is uniformly expanding or uniformly hyperbolic on an -invariant compact set if there exists and such that everywhere on . There are two main classes of (nonuniformly hyperbolic) examples we can work with. The first class is closely related to parabolic Cantor sets, introduced in .
Example 1 (parabolic IFS).
Assume that everywhere except a finite set of fixed points where . Assume also that is for some positive . We construct the subsystems by removing some small 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 : , .
Strictly speaking, the Manneville-Pomeau map is not conjugated to a subshift of finite type (some cylinders are only essentially disjoint, thus there exists a countable family of points belonging to two different cylinders of the same level). We will allow this situation, our proofs work in this case as well without major changes.
The second class is related to the one introduced in .
Example 2 (expansive Markov systems).
Consider less restrictive assumptions about , demanding only that
Assume also that is piecewise . The subsystems are constructed like in the previous case. Their hyperbolicity and bounded distortion property follows from the Mañé hyperbolicity theorem, see  for the reference.
For both the above-mentioned classes of examples we have equality in the assertion of Theorem 2.
2.2. Topological pressure
Let be a continuous function on . The topological pressure of (with respect to ) is defined by
where here and in the sequel the sum is taken over the cylinders with non-empty intersection with the set . The existence of the limit follows easily from the fact that the sum constitutes a sub-multiplicative sequence. Moreover, the value does not depend on the particular Markov partition that we use in its definition.
Denote by the family of -invariant Borel probability measures on . We simply write if there is no confusion about the system. By the variational principle we have
where denotes the entropy of with respect to (see ). A measure is called equilibrium state for the potential if
Given , we define the function by
The function is a continuous, convex, and non-increasing function of . is negative for large if and only if there exist no -invariant probability measures with zero Lyapunov exponent.
The claimed properties follow immediately from general facts about the pressure together with the variational principle (5). ∎
We have .
It follows immediately from the generator condition that every ergodic -invariant measure has non-negative Lyapunov exponent. Suppose now that for some . Then, by the variational principle, there exists an ergodic -invariant measure such that and thus . It follows then from  that
We now give a geometric description of defined in (1), (2) for positive . In general, is always a concave function with range . Let us write . Note that, by continuity and convexity, the pressure function may fail to be differentiable on an at most countable set. We sketch below the particular case that we may have at most one point of non-differentiability. If then
If for all , that is, the system is parabolic, then we have
In this situation we have the following two possible cases.
Case I: The pressure function is differentiable at . Then is strictly decreasing for positive .
Case II: The pressure function is not differentiable at . Then for every , and is strictly decreasing for greater .
2.3. Conformal measures
The Ruelle-Perron-Frobenius transfer operator defined on the space of continuous functions is given by
if . Denote by the spectral radius of . Let be an eigenmeasure of the dual operator with eigenvalue . Note that is a probability measure but not necessarily -invariant. However, the dynamical properties of with respect to are captured through its Jacobian. The Jacobian of with respect to is the (essentially) unique function determined through
for every Borel subset of such that in injective, and is given by (see ). Moreover, by [19, Theorem 2.1] we have . A measure satisfying (7) is called -conformal measure. Such a measure always exists if is expansive and open ([4, Theorem 3.12]).
3. Hyperbolic sub-systems
For shortness, we will write
Given , the function is analytic and strictly decreasing. For fixed , the sequence is non-decreasing.
Given , we have .
Assume that this is not the case for some . Clearly, form an increasing sequence. Notice that . Let . Let such that for every .
There exists a sequence of -conformal measures (with respect to the sub-system ) which we denote by . Each such measure satisfies
for every and every . Hence, from the tempered distortion property (3) we can conclude that
Notice that this inequality holds only for cylinders which intersects . However, if intersects , then it intersects for every sufficiently big.
Likewise for we obtain for the -conformal measure
for every cylinder intersecting . Hence, we obtain for every and every cylinder which intersects .
for every . Take a subsequence converging to some probability measure in the weak topology. Then we obtain
for every . This contradicts the fact that both measures are probability measures. ∎
We introduce some further notation. Let
Those are easy to calculate using the pressure, since we have
We have for
hence, by Proposition 2, we obtain
Similarly, for and big enough we have
The opposite inequalities follow from the definition of and and from . ∎
Given and let us denote
For every we have
First notice that we can rewrite
The analogous relation holds for with replaced by . Let us assume that there exists such that for every . This would imply that for every the set
is non-empty, closed, and bounded. Moreover, as , we have . Hence, is non-empty. For we conclude that
for every . Together with Proposition 2 we hence would obtain
which is a contradiction.
We mention a second way of proving (8) which is based on the convex conjugate functions. Let
denote the convex conjugate of . Then form a Legendre-Fenchel pair. Wijsman  has shown that for given Legendre-Fenchel pairs and , the functions converge infimally to if and only if converges infimally to (we refer to  for the definition of infimal convergence). In general, this kind of convergence does not coincide with the pointwise convergence. However, by monotonicity and continuity of the pressure function we obtain that converges infimally if and only if it converges pointwise. The application of Proposition 2 implies (8). ∎
For the remainder of this section let be some -invariant compact set such that is uniformly expanding. We have the following result by Jenkinson .
Lemma 4 ().
For any there exists a number and some equilibrium state for the potential (with respect to ) such that
We finally collect results on the dimension of level sets for hyperbolic systems.
For every we have
Given any Hölder continuous potential , there exists a unique ergodic equilibrium state which moreover has the Gibbs property, that is, for which there exists a constant such that for all and every we have
We refer for example to  for more details and references of the above results.
4. Upper bound for the dimension
We have for every ,
Note first that for any . Hence, the second assertion follows from the first one.
We now prove the first assertion. For a point there exists a positive number and a sequence for which we have
Let . There exists such that
for every . By the tempered distortion property (3) we obtain
Using again the tempered distortion property (3) we can conclude that the -conformal measure satisfies
and in particular the limit on the left hand side exists. Hence, possibly after increasing , we have
for every .
Thus we obtain
There exists such that, perhaps after increasing again, we have
for every . Note that . Hence, we obtain the following upper bound for the lower pointwise dimension at
Thus we obtain
for every , possibly after increasing , and hence in this case
In both cases, continuity of implies that for any given sufficiently small interval there exist such that
We can then choose a countable family of intervals , covering , and consider the corresponding sequence . Define
Since and can be chosen arbitrarily small, this finishes the proof. ∎
Given we denote
The following proposition is proved in a similar way to Proposition 4.
We have for every
5. Lower bound for the dimension
5.1. The interior of the spectrum – regular points
For we have
5.2. Construction of w-measures and their properties
Recall the notation for hyperbolic sub-systems introduced in Section 3. Given a nondecreasing sequence of positive integers , let be a sequence of certain equilibrium states for potentials with respect to . We denote
(Note that the last equality uses a result in .) We note that the same construction can be performed for an arbitrary, not necessarily non-decreasing, sequence . But this assumption simplifies the exposition. We will in the following assume that
(note that otherwise we can replace by without changing the equilibrium state ).
We now describe the construction of a measure , satisfying certain special properties. Let be a fast increasing sequence of positive integers. We will specify the specific growth speed in the course of this section. We demand that
where is a positive sequence decreasing to as in (3). We define a probability measure on the algebra generated by the cylinders . As the beginning of the construction, for cylinders of level we define
Given a cylinder of level of positive measure , we sub-distribute the measure on its sub-cylinders of level which intersect in the following way. Let
is the normalizing constant. For every