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February  2008, 22(1&2): 215-234. doi: 10.3934/dcds.2008.22.215

Infimum of the metric entropy of hyperbolic attractors with respect to the SRB measure

Huyi Hu 1, , Miaohua Jiang 2, and  Yunping Jiang 3,

1. 

Department of Mathematics, Michigan State University, East Lansing, MI 48824, United States
2. 

Department of Mathematics, Wake Forest University, Winston Salem, NC 27109
3. 

Department of Mathematics, The Graduate Center and Queens College of CUNY, Flushing, NY 11367, United States

Received  July 2007 Revised  December 2007 Published  June 2008

Let $M^{n}$ be a compact $C^{\infty}$ Riemannian manifold of dimension $n\geq 2$. Let $\text{Diff}^{\r }(M^{n})$ be the space of all $C^{\r }$ diffeomorphisms of $M^{n}$, where $ 1 < r \le \infty$. For a $C^{\r }$ diffeomorphism $f$ in $\text{Diff}^{\r }(M^{n})$ with a hyperbolic attractor $\Lambda_{f}$ on which $f$ is topologically transitive, let $U(f)$ be the $C^{1}$ open set of $\text{Diff}^{\r }(M^{n})$ such that each element in $U(f)$ can be connected to $f$ by finitely many $C^{1}$ structural stability balls in $\text{Diff}^{\r }(M^{n})$. Then by the structural stability, any element $g$ in $U(f)$ has a hyperbolic attractor $\Lambda_{g}$ and $g|\Lambda_{g}$ is topologically conjugate to $f|\Lambda_{f}$. Therefore, the topological entropy $h(g|\Lambda_{g})$ is a constant function when it is restricted to $U(f)$. However, the metric entropy $h_{\mu}(g)$ with respect to the SRB measure $\mu=\mu_{g}$ can vary. We prove that the infimum of the metric entropy $h_{\mu}(g)$ on $U(f)$ is zero.
Keywords: metric entropy, SRB measure, infimum, hyperbolic attractor..
Mathematics Subject Classification: Primary: 37C40; Secondary: 37A35, 37D2.
Citation: Huyi Hu, Miaohua Jiang, Yunping Jiang. Infimum of the metric entropy of hyperbolic attractors with respect to the SRB measure. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 215-234. doi: 10.3934/dcds.2008.22.215
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