On the hybrid control of metric entropy for dominated splittings
Xufeng Guo Gang Liao Wenxiang Sun Dawei Yang
Discrete & Continuous Dynamical Systems - A 2018, 38(10): 5011-5019 doi: 10.3934/dcds.2018219

Let $f$ be a $C^1$ diffeomorphism on a compact Riemannian manifold without boundary and $\mu$ an ergodic $f$-invariant measure whose Oseledets splitting admits domination. We give a hybrid estimate from above for the metric entropy of $\mu$ in terms of Lyapunov exponents and volume growth. Furthermore, for any $C^1$ diffeomorphism away from tangencies, its topological entropy is bounded by the volume growth.

keywords: Metric entropy Lyapunov exponent volume growth
Hyperbolic periodic points for chain hyperbolic homoclinic classes
Wenxiang Sun Yun Yang
Discrete & Continuous Dynamical Systems - A 2016, 36(7): 3911-3925 doi: 10.3934/dcds.2016.36.3911
In this paper we establish a closing property and a hyperbolic closing property for thin trapped chain hyperbolic homoclinic classes with one dimensional center in partial hyperbolicity setting. Taking advantage of theses properties, we prove that the growth rate of the number of hyperbolic periodic points is equal to the topological entropy. We also obtain that the hyperbolic periodic measures are dense in the space of invariant measures.
keywords: topological entropy growth of periodic points Chain hyperbolicity maximal entropy measures. closing property
Zero entropy versus infinite entropy
Wenxiang Sun Cheng Zhang
Discrete & Continuous Dynamical Systems - A 2011, 30(4): 1237-1242 doi: 10.3934/dcds.2011.30.1237
We construct a pair of equivalent flows with fixed points, such that one has infinite topological entropy and the other has zero topological entropy.
keywords: Topological entropy Equivalent flows.
Dominated splitting and Pesin's entropy formula
Wenxiang Sun Xueting Tian
Discrete & Continuous Dynamical Systems - A 2012, 32(4): 1421-1434 doi: 10.3934/dcds.2012.32.1421
Let $M$ be a compact manifold and $f:\,M\rightarrow M$ be a $C^1$ diffeomorphism on $M$. If $\mu$ is an $f$-invariant probability measure which is absolutely continuous relative to Lebesgue measure and for $\mu$ $a.\,\,e.\,\,x\in M,$ there is a dominated splitting $T_{orb(x)}M=E\oplus F$ on its orbit $orb(x)$, then we give an estimation through Lyapunov characteristic exponents from below in Pesin's entropy formula, i.e., the metric entropy $h_\mu(f)$ satisfies $$h_{\mu}(f)\geq\int \chi(x)d\mu,$$ where $\chi(x)=\sum_{i=1}^{dim\,F(x)}\lambda_i(x)$ and $\lambda_1(x)\geq\lambda_2(x)\geq\cdots\geq\lambda_{dim\,M}(x)$ are the Lyapunov exponents at $x$ with respect to $\mu.$
    Consequently, we obtain that Pesin's entropy formula always holds for (1) volume-preserving Anosov diffeomorphisms, (2) volume-preserving partially hyperbolic diffeomorphisms with one-dimensional center bundle, (3) volume-preserving diffeomorphisms far away from homoclinic tangency, and (4) generic volume-preserving diffeomorphisms.
keywords: Metric entropy dominated splitting. Lyapunov exponents Pesin's entropy formula
Some results on perturbations of Lyapunov exponents
Chao Liang Wenxiang Sun Jiagang Yang
Discrete & Continuous Dynamical Systems - A 2012, 32(12): 4287-4305 doi: 10.3934/dcds.2012.32.4287
In this paper, we study two properties of the Lyapunov exponents under small perturbations: one is when we can remove zero Lyapunov exponents and the other is when we can distinguish all the Lyapunov exponents. The first result shows that we can perturb all the zero integrated Lyapunov exponents $\int_M \lambda_j(x)d\omega(x)$ into nonzero ones, for any partially hyperbolic diffeomorphism. The second part contains an example which shows the local genericity of diffeomorphisms with non-simple spectrum and three results: one discusses the relation between simple-spectrum property and the existence of complex eigenvalues; the other two describe the difference on the spectrum between the diffeomorphisms far from homoclinic tangencies and those in the interior of the complement. Moreover, among the conservative diffeomorphisms far from tangencies, we prove that ergodic ones form a residual subset.
keywords: dominated splitting Integrated Lyapunov exponent ergodic. far from tangencies simple spectrum Oseledec splitting

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