## Measures of maximal entropy for surface diffeomorphisms.(English)Zbl 07483862

Decades ago, Newhouse put forward the question if a $$C^\infty$$ diffeomorphism on a closed surface with positive topological entropy has a finite number of ergodic measures of maximal entropy. This paper answers Newhouse’s question affirmatively as stated in its Main Theorem: Let $$f$$ be a $$C^\infty$$ diffeomorphism on a closed surface and suppose its topological entropy $$h_{\mathrm{top}}(f) > 0$$. Then (a) the number of ergodic measures of maximal entropy of $$f$$ is finite, (b) when $$f$$ is topologically transitive, it has a unique measure of maximal entropy, and (c) when $$f$$ is topologically mixing, it unique measures of maximal entropy is isomorphic to a Bernoulli scheme. The proof of the Main Theorem is a consequence of a generalization of Smale’s Spectral Decomposition Theorem to $$C^\infty$$ diffeomorphisms on closed surfaces with positive topological entropy, and a theorem that describes the dynamics from both the measurable and symbolic points of view, the latter being countable state Markov shifts.
The methods employed in the paper to $$C^\infty$$ diffeomorphisms and measures of maximal entropy on closed surfaces also apply to $$C^r$$ diffeomorphisms and equilibrium measures on closed surfaces for $$r>1$$. For $$M$$ a closed surface and $$f:M\to M$$ a $$C^r$$ diffeomorphism set $$\Vert Df^n\Vert = \max\{\Vert Df^n\vert_{T_xM}\Vert:x\in M\}$$, $$\lambda^u(f) = \lim_{n\to\infty}(1/n)\log \Vert Df^n\Vert$$, $$\lambda^s(f) = \lim_{n\to\infty}(1/n)\Vert Df^{-n}\Vert$$, and $$\lambda_{\min}(f) = {\min}\{\lambda^s(f),\lambda^u(f)\}$$. An extension of the Main Theorem is: Let $$f$$ be a $$C^r$$ diffeomorphism of a closed surface $$M$$ for $$r>1$$, and let $$\phi:M\to {\mathbb R}\cup\{-\infty\}$$ be an admissible potential. Then (a) for any $$\chi>\lambda_{\min}(f)/r$$ there are at most finitely many ergodic equilibrium measures for $$\phi$$ with entropy strictly bigger than $$\chi$$, and (b) each compact invariant transitive subset of $$M$$ carries at most one ergodic hyperbolic equilibrium measure $$\mu$$ for $$\phi$$ with the unstable dimension of $$\mu$$ bigger than $$1/r$$, and at most one ergodic hyperbolic equilibrium measure $$\mu$$ for $$\phi$$ with stable dimension bigger than $$1/r$$. A Corollary of this extension of the Main Theorem is: Let $$f$$ be a $$C^r$$ diffeomorphism on a closed surface for $$r>1$$, and let $$\phi$$ be an admissible potential. Assume that $\sup_{\nu\in {\mathbb P}_e(f)} \left\{ h(f,\nu) + \int \phi\ d\nu\right\} > \sup\phi + \frac{\lambda_{\min}(f)}{r}$ where $${\mathbb P}_e(f)$$ is the set of $$f$$-invariant ergodic measures and $$h(f,\mu)$$ is the metric entropy of $$f$$ with respect to $$\nu$$. Then $$f$$ has at most infinitely many ergodic equilibrium measures, and if $$f$$ is topologically transitive, then it has at most one.

### MSC:

 37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems 37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) 37D35 Thermodynamic formalism, variational principles, equilibrium states for dynamical systems 37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces 37B10 Symbolic dynamics 37B40 Topological entropy
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