Krerley Oliveira

Thermodynamical formalism for robust classes of potentials and non-uniformly hyperbolic maps

We develop a Ruelle-Perron-Frobenius transfer operator approach to the ergodic theory of a large class of non-uniformly expanding transformations on compact manifolds. For Holder continuous potentials not too far from constant, we prove that the transfer operator has a positive eigenfunction, piecewise Holder continuous, and use this fact to show that there is exactly one equilibrium state. Moreover, the equilibrium state is a nonlacunary Gibbs measure, a non-uniform version of the classical notion of Gibbs measure that we introduce here. Dedicated to the memory of William Parry

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.

Shadowing by non-uniformly hyperbolic periodic points and uniform hyperbolicity

We prove that, under a mild condition on the hyperbolicity of its periodic points, a map g which is topologically conjugated to a hyperbolic map (respectively, an expanding map) is also a hyperbolic map (respectively, an expanding map). In particular, this result gives a partial positive answer for a question asked by A Katok, in a related context.

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.

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 and Pollicott (Factors of Gibbs measures for full shifts, entropy of hidden Markov processes and connections to dynamical systems, Cambridge University Press, Cambridge, 2011) and Chazottes and Ugalde (On the preservation of Gibbsianness under symbol amalgamation, entropy of hidden Markov processes and connections to dynamical systems, Cambridge University Press, Cambridge, 2011), 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.

ON THE CONTINUITY OF ENTROPY FOR NON-UNIFORMLY EXPANDING MAPS

It is a difficult problem to verify the existence of these measures for general Dynamical Systems. For Axiom A diffeomorphisms and flows, and uniformly expanding endomorphisms classical results by Sinai, Ruelle and Bowen prove the existence of SRB measures. In this context the continuous variation of the SRB entropy is known. This follows easily from the fact that it coincides with the integral of the Jacobian with respect to