28 research outputs found
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 -dimensional immersed Riemannian manifold in some
Euclidean space we prove that if the Hausdorff dimension of the singular set of
the Gauss map is small, then is homeomorphic to the sphere .
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 -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 -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 (-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 for
a diffeomorphism assuming that it is either or is
and the Oseledec splitting of is dominated. We show that this
system 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