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

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  • 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.
    Mathematics Subject Classification: Primary: 37C40; Secondary: 37A35, 37D20.

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