Wenxiang Sun

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 Katoks 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 Liaos style number (Definition 1.3) which are bases for discussing the approximation properties in the main result.

Lyapunov exponents of hyperbolic measures and hyperbolic periodic orbits

Lyapunov exponents of a hyperbolic ergodic measure are approximated by Lyapunov exponents of hyperbolic atomic measures on periodic orbits.

Entropy of flows, revisited

We introduce a concept of measure-theoretic entropy for flows and study its invariance under measure-theoretic equivalences. Invariance properties of the corresponding topological entropy is studied too. We also answer a question posed by Bowen-Walters in [3] concerning the equality between the topological entropy of the time-one map of an expansive flow and the time-one map of its symbolic suspension.

Entropy of orthonormal n-frame flows

We introduce an equivalent definition of the entropy for flows, and by using the equivalent definition we answer a problem raised by Liao concerning Liao hyperbolic systems, i.e. the state flow and its orthonormal n-frame flow having the same entropy.

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).

Variational equalities of entropy in nonuniformly hyperbolic systems

In this paper we prove that for a nonuniformly hyperbolic system (f, ̃ Λ) and for every nonempty, compact and connected subset K with the same hyperbolic rate in the space Minv( ̃ Λ, f) of invariant measures on ̃ Λ, the metric entropy and the topological entropy of basin GK are related by the variational equality inf{hμ(f) | μ ∈ K} = htop(f,GK). In particular, for every invariant (usually nonergodic) measure μ∈Minv( ̃ Λ, f), we have hμ(f) = htop(f,Gμ). We also verify thatMinv( ̃ Λ, f) contains an open domain in the space of ergodic measures for diffeomorphisms with some hyperbolicity. As an application, the historical behavior is shown to occur robustly with a full positive entropy for diffeomorphisms beyond uniform hyperbolicity.

Topological entropy and the complete invariant for expansive maps

The topological entropy of an expansive map is equal to that of the corresponding symbolic system. The topological entropy and ergodic period are a complete invariant index (h ,b ) for an equivalence relation, almost topological conjugacy, in the setting of ergodically supported expansive maps with shadowing property, including Anosov maps.

A note on approximation properties of the Oseledets splitting

We prove that the Oseledets splitting, mean angle and independence number of an ergodic hyperbolic measure of a C1+r diffeomorphism can be approximated by those of atomic measures on hyperbolic periodic orbits. This removes the assumption on simple spectrum in an earlier paper by the authors and strengthens Katok’s closing lemma by presenting more information about not only the state space but also its linearization.

Non-uniformly hyperbolic periodic points and uniform hyperbolicity for C1 diffeomorphisms

We prove that for C1 generic diffeomorphisms, every isolated compact invariant set Λ which satisfies a mild condition on the hyperbolicity of periodic points in Λ (called the L-NUH condition, see definition 1.1) is hyperbolic. In parallel, we prove that for C1 diffeomorphisms, every compact invariant set which satisfies Katoks periodic closing property and the L-NUH condition on periodic points is hyperbolic, which is a generalized result of Castro et al (2007 Nonlinearity 20 75-85) for C2 case with a periodic closing property (called periodic shadowing property in Castro et al (2007 Nonlinearity 20 75-85)).