Ergodic Hyperbolic Measure Research Articles

Horseshoes and Lyapunov exponents for Banach cocycles over non-uniformly hyperbolic systems

AbstractWe extend Katok’s result on ‘the approximation of hyperbolic measures by horseshoes’ to Banach cocycles. More precisely, let f be a $C^r(r>1)$ diffeomorphism of a compact Riemannian manifold M, preserving an ergodic hyperbolic measure $\mu $ with positive entropy, and let $\mathcal {A}$ be a Hölder continuous cocycle of bounded linear operators acting on a Banach space $\mathfrak {X}$ . We prove that there is a sequence of horseshoes for f and dominated splittings for $\mathcal {A}$ on the horseshoes, such that not only the measure theoretic entropy of f but also the Lyapunov exponents of $\mathcal {A}$ with respect to $\mu $ can be approximated by the topological entropy of f and the Lyapunov exponents of $\mathcal {A}$ on the horseshoes, respectively. As an application, we show the continuity of sub-additive topological pressure for Banach cocycles.

The approximation of uniform hyperbolicity for C1 diffeomorphisms with hyperbolic measures

Let f be a C1-diffeomorphism which preserves an ergodic hyperbolic measure μ with positive measure-theoretic entropy. Assume that the Oseledec splittingTxM=E1(x)⊕⋯⊕Es(x)⊕Es+1(x)⊕⋯⊕El(x) is dominated on the Oseledec basin Γ. We give extensions of Katok's Horseshoes construction. Moreover there is a dominated splitting corresponding to Oseledec subspace on horseshoes. Based on this, we also establish the continuity of the sub-additive topological pressure of the singular value potential.

Pressure and exponential rate over periodic orbits

For a measure μ preserved by a C1+α0 (α0>0) diffeomorphism f and a continuous function ϕ on the manifold, we study the relationship between the exponential growth rate of ∑eSnϕ(x) over the orbits of some periodic points and the free energy hμ(f)+∫ϕdμ (in certain cases, it is equal to the measure theoretic pressure) or the topological pressure. When μ is an ergodic hyperbolic measure, we prove that the exponential growth rate coincides with the free energy (measure theoretic pressure). And we also verify the equality of the exponential growth rate and the topological pressure when the manifold is 2-dimensional. However, for the higher-dimensional manifold, we show an inequality between the exponential growth rate and the topological pressure. For an ergodic hyperbolic measure ω, we also prove that there is a ω-full measured set Λ˜ such that for every f-invariant measure supported on Λ˜, the exponential growth rate equals to the free energy. And moreover, we prove that there is another ω-full measured set Δ˜ such that for every f-invariant measure supported on Δ˜, the exponential growth rate equals to the topological pressure.

Invariant multi-graphs in step skew-products

ABSTRACTWe study step skew-products over a finite-state shift (base) space whose fibre maps are injective maps on the unit interval. We show that certain invariant sets have a multi-graph structure and can be written graphs of one, two or more functions defined on the base. In particular, this applies to any hyperbolic set and to the support of any ergodic hyperbolic measure. Moreover, within the class of step skew-products whose interval maps are ‘absorbing’, open and densely the phase space decomposes into attracting and repelling double-strips such that their attractors and repellers are graphs of one single-valued or bi-valued continuous function almost everywhere, respectively.

Livšic theorem for matrix cocycles over nonuniformly hyperbolic systems

We prove a nonuniformly hyperbolic version of the Livšic-type theorem, with cocycles taking values in [Formula: see text]. To be more precise, let [Formula: see text] Diff[Formula: see text] preserving an ergodic hyperbolic measure [Formula: see text], and [Formula: see text] be Hölder continuous satisfying [Formula: see text] for each periodic point [Formula: see text], then there exists a measurable function [Formula: see text] satisfying [Formula: see text] for [Formula: see text]-almost every [Formula: see text].

Exceptional Sets for Nonuniformly Hyperbolic Diffeomorphisms

For a surface diffeomorphism, a compact invariant locally maximal set W and some subset $$A\subset W$$ we study the A-exceptional set, that is, the set of points whose orbits do not accumulate at A. We show that if the Hausdorff dimension of A is smaller than the Hausdorff dimension d of some ergodic hyperbolic measure, then the topological entropy of the exceptional set is at least the entropy of this measure and its Hausdorff dimension is at least d. Particular consequences occur when there is some a priori defined hyperbolic structure on W and, for example, if there exists an SRB measure.

On the approximation of dynamical indicators in systems with nonuniformly hyperbolic behavior

Let f be a \(C^{1+\alpha }\) diffeomorphism of a compact Riemannian manifold and \(\mu \) an ergodic hyperbolic measure with positive entropy. We prove that for every continuous potential \(\phi \) there exists a sequence of basic sets \(\Omega _n\) such that the topological pressure \(P(f|\Omega _n,\phi )\) converges to the free energy \(P_{\mu }(\phi ) = h(\mu ) + \int \phi {d\mu }\). We also prove that for a suitable class of potentials \(\phi \) there exists a sequence of basic sets \(\Omega _n\) such that \(P(f|\Omega _n,\phi ) \rightarrow P(\phi )\).

Exponential rate of periodic points and metric entropy in nonuniformly hyperbolic systems

For an ergodic hyperbolic measure ω of a C1+r diffeomorphism f there is an ω-full measured set Λ˜ such that for every invariant measure μ∈Minv(Λ˜,f) the exponential growth rate of the number of periodic points whose atomic measures approximate μ equals to the metric entropy hμ(f).

The Hausdorff dimension estimation for an ergodic hyperbolic measure of $C^1$-diffeomorphism

The Hausdorff dimension estimation for an ergodic hyperbolic measure of $C^1$-diffeomorphism

The distribution of the periodic orbits with open deviation in C 1+α non-uniformly hyperbolic systems

Let f: M → M be a C 1+α diffeomorphism on a smooth compact Riemannian manifold M and Λ be a Pesin set associated with the ergodic hyperbolic measure µ. Then f: Λ → Λ forms a non-uniformly hyperbolic system. We concern with the distribution of the periodic orbits whose time averages are apart from the space average of µ. Finally, we derive a large deviation result for these periodic orbits with open deviation property.

Horseshoes for $\mathcal{C}^{1+\alpha}$ mappings with hyperbolic measures

We present here a construction of horseshoes for any $\mathcal{C}^{1+\alpha}$ mapping $f$ preserving an ergodic hyperbolic measure $\mu$ with $h_{\mu}(f)>0$ and then deduce that the exponential growth rate of the number of periodic points for any $\mathcal{C}^{1+\alpha}$ mapping $f$ is greater than or equal to $h_{\mu}(f)$. We also prove that the exponential growth rate of the number of hyperbolic periodic points is equal to the hyperbolic entropy. The hyperbolic entropy means the entropy resulting from hyperbolic measures.

Non-uniform hyperbolicity and non-uniform specification

In this paper we deal with an invariant ergodic hyperbolic measure $\mu$ for a diffeomorphism $f,$ assuming that $f$ it is either $C^{1+\alpha}$ or $f$ is $C^1$ and the Oseledec splitting of $\mu$ is dominated. We show that this system $(f,\mu)$ satisfies a weaker and non-uniform version of specification, related with notions studied in several recent papers, including \cite{STV,Y, PS, T,Var, Oli}. Our main results have several consequences: as corollaries, we are able to improve the results about quantitative Poincar\'e recurrence, removing the assumption of the non-uniform specification property in the main Theorem of \cite{STV} that establishes an inequality between Lyapunov exponents and local recurrence properties. Another consequence is the fact that any of such measure is the weak limit of averages of Dirac measures at periodic points, as in \cite{Sigmund}. Following \cite{Y} and \cite{PS}, one can show that the topological pressure can be calculated by considering the convenient weighted sums on periodic points, whenever the dynamics is positive expansive and every measure with pressure close to the topological pressure is hyperbolic.

Metric entropy and the number of periodic points

For an ergodic hyperbolic measure μ preserved by a C1+r(r > 0) diffeomorphism f, the exponential growth rate of the number of such periodic points that their atomic measures approximate μ and their Lyapunov exponents approximate the Lyapunov exponents of μ equals the metric entropy hμ(f) (see theorem 2.3). Moreover, this equality holds pointwise μ-a.e. (see theorem 2.4).

Dimension Theory for Invariant Measures of Endomorphisms

We establish the exact dimensional property of an ergodic hyperbolic measure for a C 2 non-invertible but non-degenerate endomorphism on a compact Riemannian manifold without boundary. Based on this, we give a new formula of Lyapunov dimension of ergodic measures and show it coincides with the dimension of hyperbolic ergodic measures in a setting of random endomorphisms. Our results extend several well known theorems of Barreira et al. (Ann Math 149:755–783, 1999) and Ledrappier and Young [Commun Math Phys 117(4):529–548, 1988] for diffeomorphisms to the case of endomorphisms.

Approximation properties on invariant measure and Oseledec splitting in non-uniformly hyperbolic systems

We prove that each invariant measure in a non-uniformly hyperbolic system can be approximated by atomic measures on hyperbolic periodic orbits. This contributes to our main result that the mean angle (Definition 1.10), independence number (Definition 1.6) and Oseledec splitting for an ergodic hyperbolic measure with simple spectrum can be approximated by those for atomic measures on hyperbolic periodic orbits, respectively. Combining this result with the approximation property of Lyapunov exponents by Wang and Sun, 2005 (Theorem 1.9), we strengthen Katok's closing lemma (1980) by presenting more extensive information not only about the state system but also its linearization. In the present paper, we also study an ergodic theorem and a variational principle for mean angle, independence number and Liao's style number (Definition 1.3) which are bases for discussing the approximation properties in the main result.

On hyperbolic measures and periodic orbits

We prove that if a diffeomorphism on a compact manifold preserves a nonatomic ergodic hyperbolic Borel probability measure, then there exists a hyperbolic periodic point such that the closure of its unstable manifold has positive measure. Moreover, the support of the measure is contained in the closure of all such hyperbolic periodic points. We also show that if an ergodic hyperbolic probability measure does not locally maximize entropy in the space of invariant ergodic hyperbolic measures, then there exist hyperbolic periodic points that satisfy a multiplicative asymptotic growth and are uniformly distributed with respect to this measure.

Shadowing Property of Non-Invertible Maps with Hyperbolic Measures

We show that if a differentiable map of a smooth manifold has a non-atomic ergodic hyperbolic measure then the topological entropy is positive and the space contains a hyperbolic horseshoe. Moreover we give some relations between hyperbolic measures and periodic points for differentiable maps. These are generalized contents of the results obtained by Katok for diffeomorphisms.