Every expanding measure has the nonuniform specification property

Exploring abundance and non lacunarity of hyperbolic times for endomorphisms preserving an ergodic probability with positive Lyapunov exponents, we obtain that there are periodic points of period growing sublinearly with respect to the lenght of almost every dynamical ball. In particular, we conclude that any ergodic measure with positive Lyapunov exponents satisfy the nonuniform specification property. As consequences, we (re)-obtain estimates on the recurrence to a ball in terms of the Lyapunov exponents and we prove that any expanding measure is limit of Dirac measures on periodic points.Comment: 10 page

Equilibrium States for Non-uniformly expanding maps

We construct equilibrium states, including measures of maximal entropy, for a large (open) class of non-uniformly expanding maps on compact manifolds. Moreover, we study uniqueness of these equilibrium states, as well as some of their ergodic properties.Comment: 18 page

Geometrical versus Topological Properties of Manifolds

Given a compact $n$-dimensional immersed Riemannian manifold $M^n$ in some Euclidean space we prove that if the Hausdorff dimension of the singular set of the Gauss map is small, then $M^n$ is homeomorphic to the sphere $S^n$. Also, we define a concept of finite geometrical type and prove that finite geometrical type hypersurfaces with small set of points of zero Gauss-Kronecker curvature are topologically the sphere minus a finite number of points. A characterization of the $2n$-catenoid is obtained

An optimization problem with free boundary governed by a degenerate quasilinear operator

In this paper we study the existence, regularity and geometric properties of an optimal configuration to a free boundary optimization problem governed by the $p$-Laplacian.Comment: 16 page

Sequential Gibbs Measures and Factor Maps

We define the notion of sequential Gibbs measures, inspired by on the classical notion of Gibbs measures and recent examples from the study of non-uniform hyperbolic dynamics. Extending previous results of Kempton-Pollicott and Ugalde-Chazottes, we show that the images of one block factor maps of a sequential Gibbs measure are also a sequential Gibbs measure, with the same sequence of Gibbs times. We obtain some estimates on the regularity of the potential of the image measure at almost every point.Comment: To appear Journal of Statistical Physic

Equilibrium states for natural extensions of non-uniformly expanding local homeomorphisms

We examine uniqueness of equilibrium states for the natural extension of a topologically exact, non-uniformly expanding, local homeomorphism with a H\"older continuous potential function. We do this by applying general techniques developed by Climenhaga and Thompson, and show there is a natural condition on decompositions that guarantees that a unique equilibrium state exists. We then show how to apply these results to partially hyperbolic attractors.Comment: 12 pages. arXiv admin note: text overlap with arXiv:1909.0523

Existence and uniqueness of maximizing measures for robust classes of local diffeomorphisms

We prove existence of maximal entropy measures for an open set of non-uniformly expanding local diffeomorphisms on a compact Riemannian manifold. In this context the topological entropy coincides with the logarithm of the degree, and these maximizing measures are eigenmeasures of the transfer operator. When the map is topologically mixing, the maximizing measure is unique and positive on every open set.Comment: 15 page

Equilibrium States for Random Non-uniformly Expanding Maps

We show that, for a robust ($C^2$-open) class of random non-uniformly expanding maps, there exists equilibrium states for a large class of potentials.In particular, these sytems have measures of maximal entropy. These results also give a partial answer to a question posed by Liu-Zhao. The proof of the main result uses an extension of techniques in recent works by Alves-Ara\'ujo, Alves-Bonatti-Viana and Oliveira

Non-uniform Hyperbolicity and Non-uniform Specification

In this paper we deal with an invariant ergodic hyperbolic measure $\mu$ for a diffeomorphism $f,$ assuming that $f$ it is either $C^{1+\alpha}$ or $f$ is $C^1$ and the Oseledec splitting of $\mu$ is dominated. We show that this system $(f,\mu)$ satisfies a weaker and non-uniform version of specification, related with notions studied in several recent papers, including \cite{STV,Y, PS, T,Var, Oli}. Our main results have several consequences: as corollaries, we are able to improve the results about quantitative Poincar\'e recurrence, removing the assumption of the non-uniform specification property in the main Theorem of \cite{STV} that establishes an inequality between Lyapunov exponents and local recurrence properties. Another consequence is the fact that any of such measure is the weak limit of averages of Dirac measures at periodic points, as in \cite{Sigmund}. Following \cite{Y} and \cite{PS}, one can show that the topological pressure can be calculated by considering the convenient weighted sums on periodic points, whenever the dynamics is positive expansive and every measure with pressure close to the topological pressure is hyperbolic.Comment: 21pages, nonuniform specificatio

On the continuity of the SRB entropy for endomorphisms

We consider classes of dynamical systems admitting Markov induced maps. Under general assumptions, which in particular guarantee the existence of SRB measures, we prove that the entropy of the SRB measure varies continuously with the dynamics. We apply our result to a vast class of non-uniformly expanding maps of a compact manifold and prove the continuity of the entropy of the SRB measure. In particular, we show that the SRB entropy of Viana maps varies continuously with the map.Comment: 19 page