We show that $C^\infty$-surface diffeomorphisms with positive topological entropy have finitely many ergodic measures of maximal in general, and exactly one the topologically transitive case. This answers a question Newhouse, who proved such always exist. To do this we generalize Smale's spectral decomposition theorem to non-uniformly hyperbolic surface diffeomorphisms, introduce homoclinic cla...

There have been many attempts to understand the statistical origin of black-hole entropy. Among them, entanglement entropy and the brick wall model are strong candidates. In this paper we show a relation between entanglement entropy and the brick wall model: the brick wall model seeks the maximal value of the entanglement entropy. In other words, the entanglement approach reduces to the brick w...

Bowen showed that a continuous expansive map with specification has a unique measure of maximal entropy. We show that the conclusion remains true under weaker non-uniform versions of these hypotheses. To this end, we introduce the notions of obstructions to expansivity and specification, and show that if the entropy of such obstructions is smaller than the topological entropy of the map, then t...

It is known that piecewise affine surface homeomorphisms always have measures of maximal entropy. This is easily seen to fail in the discontinuous case. Here we describe a piecewise affine, globally continuous surface map with no measure of maximal entropy.

In this note we construct measures of maximal entropy for a certain class of maps with critical points. The main application of our result is the existence of measures of maximal entropy for the so-called Viana maps.

We define a new notion of entropy for operators on Fock spaces and positive definite multi-Toeplitz kernels on free semigroups. This is studied in connection with factorization theorems for (multi-Toeplitz, multi-analytic, etc.) operators on Fock spaces. These results lead to entropy inequalities and entropy formulas for positive definite multi-Toeplitz kernels on free semigroups (resp. multi-T...