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Jose F. Alves

Publications and source records attributed to Jose F. Alves.

34 records · Page 2Linked to original sources
arXiv2012

Statistical stability and limit laws for Rovella maps

We consider the family of one-dimensional maps arising from the contracting Lorenz attractors studied by Rovella. Benedicks-Carleson techniques were used by Rovella to prove that there is a one-parameter family of maps whose derivatives along their critical orbits increase exponentially fast and the critical orbits have slow recurrent to the critical point. Metzger proved that these maps have a unique absolutely continuous ergodic invariant probability measure (SRB measure). Here we use the technique developed by Freitas and show that the tail set (the set of points which at a given time have not achieved either the exponential growth of derivative or the slow recurrence) decays exponentially fast as time passes. As a consequence, we obtain the continuous variation of the densities of the SRB measures and associated metric entropies with the parameter. Our main result also implies some statistical properties for these maps.

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arXiv2011

Explosion of smoothness for conjugacies between multimodal maps

Let $f$ and $g$ be smooth multimodal maps with no periodic attractors and no neutral points. If a topological conjugacy $h$ between $f$ and $g$ is $C^{1}$ at a point in the nearby expanding set of $f$, then $h$ is a smooth diffeomorphism in the basin of attraction of a renormalization interval of $f$. In particular, if $f:I \to I$ and $g:J \to J$ are $C^r$ unimodal maps and $h$ is $C^{1}$ at a boundary of $I$ then $h$ is $C^r$ in $I$.

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arXiv2010

Strong stochastic stability for non-uniformly expanding maps

We consider random perturbations of discrete-time dynamical systems. We give sufficient conditions for the stochastic stability of certain classes of maps, in a strong sense. This improves the main result in J. F. Alves, V. Araujo, Random perturbations of non-uniformly expanding maps, Asterisque 286 (2003), 25--62, where it was proved the convergence of the stationary measures of the random process to the SRB measure of the initial system in the weak* topology. Here, under slightly weaker assumptions on the random perturbations, we obtain a stronger version of stochastic stability: convergence of the densities of the stationary measures to the density of the SRB measure of the unperturbed system in the L1-norm. As an application of our results we obtain strong stochastic stability for two classes of non-uniformly expanding maps. The first one is an open class of local diffeomorphisms introduced in J. F. Alves, C. Bonatti, M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly expanding, Invent. Math. 140 (2000), 351--398, and the second one the class of Viana maps.

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arXiv2009

Gibbs-Markov structures and limit laws for partially hyperbolic attractors with mostly expanding central direction

We consider a partially hyperbolic set $K$ on a Riemannian manifold $M$ whose tangent space splits as $T_K M=E^{cu}\oplus E^{s}$, for which the centre-unstable direction $E^{cu}$ expands non-uniformly on some local unstable disk. We show that under these assumptions $f$ induces a Gibbs-Markov structure. Moreover, the decay of the return time function can be controlled in terms of the time typical points need to achieve some uniform expanding behavior in the centre-unstable direction. As an application of the main result we obtain certain rates for decay of correlations, large deviations, an almost sure invariance principle and the validity of the Central Limit Theorem.

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arXiv2007

Stochastic stability of diffeomorphisms with dominated splitting

We prove that the statistical properties of random perturbations of a nonuniformly hyperbolic diffeomorphism are described by a finite number of stationary measures. We also give necessary and sufficient conditions for the stochastic stability of such dynamical systems. We show that a certain $C^2$-open class of nonuniformly hyperbolic diffeomorphisms introduced in [Alves, J; Bonatti, C. and Viana, V., SRB measures for partially hyperbolic systems with mostly expanding central direction, Invent. Math., 140 (2000), 351-398] are stochastically stable. Our setting encompasses that of partially hyperbolic diffeomorphisms as well. Moreover, the techniques used enable us to obtain SRB measures in this setting through zero-noise limit measures.

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arXiv2006

Recurrence times and rates of mixing for invertible dynamical systems

We consider invertible discrete-time dynamical systems having a hyperbolic product structure in some region of the phase space with infinitely many branches and variable recurrence time. We show that the decay of correlations of the SRB measure associated to that hyperbolic structure is related to the tail of the recurrence times. We also give sufficient conditions for the validity of the Central Limit Theorem. This generalizes previous results by Benedicks and Young.

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arXiv2006

Topological structure of (partially) hyperbolic sets with positive volume

We consider both hyperbolic sets and partially hyperbolic sets attracting a set of points with positive volume in a Riemannian manifold. We obtain several results on the topological structure of such sets for diffeomorphisms whose differentiability is bigger than one. We show in particular that there are no partially hyperbolic horseshoes with positive volume for such diffeomorphisms. We also give a description of the limit set of almost every point belonging to a hyperbolic set or a partially hyperbolic set with positive volume.

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arXiv2004

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.

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arXiv2003

Hyperbolic times: frequency vs. integrability

We consider dynamical systems on compact manifolds, which are local diffeomorphisms outside an exceptional set (a compact submanifold). We are interested in analyzing the relation between the integrability (with respect to Lebesgue measure) of the first hyperbolic time map and the existence of positive frequency of hyperbolic times. We show that some (strong) integrability of the first hyperbolic time map implies positive frequency of hyperbolic times. We also present an example of a map with positive frequency of hyperbolic times at Lebesgue almost every point but whose first hyperbolic time map is not integrable with respect to the Lebesgue measure.

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arXiv2003

Strong statistical stability of non-uniformly expanding maps

We consider families of transformations in multidimensional Riemannian manifolds with non-uniformly expanding behavior. We give sufficient conditions for the continuous variation (in the $L^1$-norm) of the densities of absolutely continuous (with respect to the Lebesgue measure) invariant probability measures for those transformations.

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arXiv2001

On the uniform hyperbolicity of some nonuniformly hyperbolic systems

We give sufficient conditions for the uniform hyperbolicity of certain nonuniformly hyperbolic dynamical systems. In particular, we show that local diffeomorphisms that are nonuniformly expanding on sets of total probability are necessarily uniformly expanding. We also present a version of this result for diffeomorphisms with nonuniformly hyperbolic sets.

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arXiv2000

Statistical stability for robust classes of maps with non-uniform expansion

We consider open sets of maps in a manifold $M$ exhibiting non-uniform expanding behaviour in some domain $S\subset M$. Assuming that there is a forward invariant region containing $S$ where each map has a unique SRB measure, we prove that under general uniformity conditions, the SRB measure varies continuously in the $L^1$-norm with the map. As a main application we show that the open class of maps introduced in [V] fits to this situation, thus proving that the SRB measures constructed in [A] vary continuously with the map.

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arXiv2000

Random perturbations of non-uniformly expanding maps

We give both sufficient conditions and necessary conditions for the stochastic stability of non-uniformly expanding maps either with or without critical sets. We also show that the number of probability measures describing the statistical asymptotic behaviour of random orbits is bounded by the number of SRB measures if the noise level is small enough. As an application of these results we prove the stochastic stability of certain classes of non-uniformly expanding maps introduced in \cite{V} and \cite{ABV}.

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