Finiteness of measures of maximal entropy for smooth saddle surface endomorphisms

TL;DR

The paper proves that for a $C^infty$ local diffeomorphism on a closed surface with $h_{top}(f)>log\deg(f)$, there are only finitely many ergodic measures of maximal entropy. The authors develop two Markov-type families of $us$-rectangles and leverage Pesin theory, Yomdin theory, and a refined analysis of unstable and stable intersections, to show that high-entropy measures must be hyperbolic of saddle type and must be homoclinically related to a finite set of saddle periodic orbits. By coding the dynamics inside homoclinic classes and invoking a dynamical Sard lemma, they bound the number of classes carrying m.m.e. and apply Katok’s criterion to deduce finiteness of ergodic m.m.e. The work also extends to non-singular attractors and discusses regularity, entropy thresholds, and potential generalizations, highlighting a substantial advance in non-invertible smooth dynamics on surfaces. Overall, the results connect entropy, hyperbolicity, and homoclinic structure to yield a robust finiteness phenomenon for measures of maximal entropy in endomorphic settings.

Abstract

We show that $\mathcal{C}^{\infty}$ local diffeomorphisms of closed surfaces whose topological entropy is larger than the logarithm of their degree admit a finite number of ergodic measures of maximal entropy. To do this, we construct families of rectangles, with a nice geometry, displaying a Markov property. We then analyze the behavior of the iterates of unstable curves intersecting these rectangles, using Yomdin theory.

Figures (10)

• Figure 1: On the left, an example of an intersection between $\tilde{R}_j$ and $f^n(\tilde{R}_i)$ which does not satisfy Property \ref{['property:MR5']}. On the right, an example of an intersection satisfying \ref{['property:MR5']}.
• Figure 2: The construction of the $us$-rectangle $\tilde{R}_2$.
• Figure 3: The estimation of $\sigma^{u,\alpha}_{\max}(\tilde{R}_2)$ in Case 1. In blue, the rectangle $f^n(R_1)$. In gray, the rectangle $R_2$. In red, the geodesic curve $c_{\delta}$. In green, the curves $c^u_1,c^u_2,c^s$ used to estimate the length of $c_{\delta}$.
• Figure 4: The estimation of $\sigma^{u,\alpha}_{\max}(\tilde{R}_2)$ in Case 2. In blue, the rectangle $f^n(R_1)$. In gray, the rectangle $R_2$. In red, the geodesic curve $c$. In green, the curves $c^u_1,c^u_2,c^s_1,c^s_2$ used to estimate the length of $c$. $c^s$ is the whole horizontal green curve while $c^s_1$ and $c^s_2$ are the subcurves going from $V^u_{2,1}$ to $z$ and $z$ to $V^u_{2,2}$ in Case 2a, or $\tilde{W}^u_{1,2}$ in Case 2b.
• Figure 5: The construction of the first family of rectangles: in light blue, the foliation $\mathscr{F}^u$; in dark blue, the $us$-rectangle.

Theorems & Definitions (134)

• Theorem A
• Theorem B
• Remark
• Theorem C
• Remark
• Corollary
• Proposition 2.1: Ruelle inequality
• Definition 2.2
• Definition 2.3
• Lemma 2.4