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
Minimal non-hyperbolicity and index-completeness
Dawei Yang Shaobo Gan Lan Wen
Discrete & Continuous Dynamical Systems - A 2009, 25(4): 1349-1366 doi: 10.3934/dcds.2009.25.1349
We study a problem raised by Abdenur et. al. [3] that asks, for any chain transitive set $\Lambda$ of a generic diffeomorphism $f$, whether the set $I(\Lambda)$ of indices of hyperbolic periodic orbits that approach $\Lambda$ in the Hausdorff metric must be an "interval", i.e., whether $\alpha\in I(\Lambda)$ and $\beta\in I(\Lambda)$, $\alpha<\beta$, must imply $\gamma\in I(\Lambda)$ for every $\alpha<\gamma<\beta$. We prove this is indeed the case if, in addition, $f$ is $C^1$ away from homoclinic tangencies and if $\Lambda$ is a minimally non-hyperbolic set.
keywords: Dimension theory multifractal analysis. Poincaré recurrences
On the hyperbolicity of homoclinic classes
Christian Bonatti Shaobo Gan Dawei Yang
Discrete & Continuous Dynamical Systems - A 2009, 25(4): 1143-1162 doi: 10.3934/dcds.2009.25.1143
We give a sufficient criterion for the hyperbolicity of a homoclinic class. More precisely, if the homoclinic class $H(p)$ admits a partially hyperbolic splitting $T_{H(p)}M=E^s\oplus_{_<}F$, where $E^s$ is uniformly contracting and $\dim E^s= \ $ind$(p)$, and all periodic points homoclinically related with $p$ are uniformly $E^u$-expanding at the period, then $H(p)$ is hyperbolic. We also give some consequences of this result.
keywords: hyperbolic time shadowing lemma. homoclinic class

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