Séance Séminaire

Séminaire Gaston Darboux

Measures of maximal entropy for saddle surfaces endomorphisms

The topological entropy of a dynamical system provides a quantitative measure of its chaotic behavior. In order to describe the statistical properties of chaotic orbits, it is often natural to study invariant ergodic probability measures for which these orbits are typical. More precisely, we are interested in measures of maximal entropy, that is, invariant measures whose metric entropy coincides with the topological entropy of the system. A first fundamental question concerns the existence and finiteness of such measures. In the smooth setting, Newhouse showed that ergodic measures of maximal entropy always exist. More recently, Buzzi, Crovisier, and Sarig proved that for smooth surface diffeomorphisms with positive entropy, the number of such measures is finite. For surface endomorphisms, however, the situation is markedly different and the behavior can be much wilder. After presenting examples where the number of measures of maximal entropy is infinite, we show that, under suitable hypotheses relating the entropy and the degree of the map, smooth surface endomorphisms admit only finitely many ergodic measures of maximal entropy. Finally, we give an overview of the main ideas of the proof and the tools involved, which are of a topological nature and allow one to control the entropy along unstable curves.