Related papers: Generic hyperbolic Ma\~n\'e sets have zero entropy

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We show that systems with some specification properties are topologically or almost Borel universal, in the sense that any aperiodic subshift with lower entropy may be topologically or almost Borel embedded.

We show that diffeomorphisms with a dominated splitting of the form $E^s\oplus E^c\oplus E^u$, where $E^c$ is a nonhyperbolic central bundle that splits in a dominated way into 1-dimensional subbundles, are entropy-expansive. In particular,…

An attractor $\Lambda$ for a 3-vector field $X$ is singular-hyperbolic if all its singularities are hyperbolic and it is partially hyperbolic with volume expanding central direction. We prove that $C^{1+\alpha}$ singular-hyperbolic…

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We give explicit $C^1$-open conditions that ensure that a diffeomorphism possesses a nonhyperbolic ergodic measure with positive entropy. Actually, our criterion provides the existence of a partially hyperbolic compact set with…

Notions of topological free entropy and of free capacity are introduced in the $C^*$-algebra context. Basic properties, basic problems and connections to potential theory and random matrix theory are discussed.

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In this paper we study two properties related to the structure of hyperbolic sets. First we construct new examples answering in the negative the following question posed by Katok and Hasselblatt. Let $\Lambda$ be a hyperbolic set, and let…

We establish multiple recurrence and convergence results for pairs of zero entropy measure preserving transformations that do not satisfy any commutativity assumptions. Our results cover the case where the iterates of the two…

We prove the following boundary-theoretic characterization of relatively hyperbolic groups. Let $G$ be a finitely generated group with a finite collection $\mathcal{H}$ of finitely generated subgroups, and let $G^h$ denote the associated…

Consider a topologically exact $C^3$ interval map without non-flat critical points. Following the works we did in \cite{LiRiv12two}, we give two equivalent characterizations of hyperbolic H\"{o}lder continuous potential in terms of the…

In this work we will show that there exists a fundamental difference between microscopic quantum thermodynamics and macroscopic classical thermodynamics. It will be proved that the entropy production in quantum thermodynamics always…

In this paper we show that any ergodic measure preserving transformation of a standard probability space which is AT$(n)$ for some positive integer $n$ has zero entropy. We show that for every positive integer $n$ any Bernoulli shift is not…