José Ferreira Alves

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• PhD
• Full Professor at University of Porto

We consider non-uniformly expanding maps on compact Riemannian manifolds of arbitrary dimension, possibly having discontinuities and/or critical sets, and show that under some general conditions they admit an induced Markov tower structure. Moreover, the decay of the return time function can be controlled in terms of the time generic points need to...

We construct ergodic absolutely continuous invariant probability measures for an open class of non-hyperbolic surface maps introduced by [V2], who showed that they exhibit two positive Lyapunov exponents at almost every point. Our approach involves an inducing procedure, based on the notion of hyperbolic time that we introduce here, and contains a...

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 enou...

We consider partially hyperbolic \( C^{1+} \) diffeomorphisms of compact Riemannian manifolds of arbitrary dimension which admit a partially hyperbolic tangent bundle decomposition \( E^s\oplus E^{cu} \). Assuming the existence of a set of positive Lebesgue measure on which \( f \) satisfies a weak nonuniform expansivity assumption in the centre~un...

We construct Sinai-Ruelle-Bowen (SRB) measures supported on partially hyperbolic sets of diffeomorphisms - the tangent bundle splits into two invariant subbundles, one of which is uniformly contracting - under the assumption that the complementary subbundle is non-uniformly expanding. If the rate of expansion (Lyapunov exponents) is bounded away fr...

In a context of non-uniformly expanding maps, possibly with the presence of a critical set, we prove the existence of finitely many ergodic equilibrium states for hyperbolic potentials. Moreover, the equilibrium states are expanding measures. This generalizes a result due to Ramos and Viana, where analytical methods are used for maps with no critic...

We study semiflows generated via impulsive perturbations of Lorenz flows. We prove that such semiflows admit a finite number of physical measures. Moreover, if the impulsive perturbation is small enough, we show that the physical measures of the semiflows are close, in the weak* topology, to the unique physical measure of the Lorenz flow. A similar...

In this article we study random tower maps driven by an ergodic automorphism. We prove quenched exponential correlations decay for tower maps admitting exponential tails. Our technique is based on constructing suitable cones of functions, defined on the random towers, which contract with respect to the Hilbert metric under the action of appropriate...

We prove strong statistical stability of a large class of one-dimensional maps which may have an arbitrary finite number of discontinuities and of non-degenerate critical points and/or singular points with infinite derivative, and satisfy some expansivity and bounded recurrence conditions. This generalizes known results for maps with critical point...

In this article we study random tower maps driven by an ergodic automorphism. We prove quenched exponential correlations decay for tower maps admitting exponential tails. Our technique is based on constructing suitable cones of functions, defined on the random towers, which contract with respect to the Hilbert metric under the action of appropriate...

We introduce random towers to study almost sure rates of correlation decay for random partially hyperbolic attractors. Using this framework, we obtain abstract results on almost sure exponential, stretched exponential and polynomial correlation decay rates. We then apply our results to small random perturbations of Axiom A attractors, small perturb...

We obtain entropy formulas for SRB measures with finite entropy given by inducing schemes. In a first part, we deduce the entropy formula for a class of systems whose SRB measures are given by Gibbs-Markov induced maps. In a second part, the entropy formula is derived for SRB measures given by Young sets, taking into account a classical compression...

We consider random perturbations of a topologically transitive local diffeomorphism of a Riemannian manifold. We show that if an absolutely continuous ergodic stationary measures is expanding (all Lyapunov exponents positive), then there is a random Gibbs–Markov–Young structure which can be used to lift that measure. We also prove that if the origi...

This monograph offers a coherent, self-contained account of the theory of Sinai–Ruelle–Bowen measures and decay of correlations for nonuniformly hyperbolic dynamical systems. A central topic in the statistical theory of dynamical systems, the book in particular provides a detailed exposition of the theory developed by L.-S. Young for systems admitt...

We consider some classes of piecewise expanding maps in finite dimensional spaces having invariant probability measures which are absolutely continuous with respect to Lebesgue measure. We derive an entropy formula for such measures and, using this entropy formula, we present sufficient conditions for the continuity of that entropy with respect to...

In the context of non-uniformly expanding maps, possibly with the presence of a critical set, we prove the existence of finitely many ergodic equilibrium states for hyperbolic potentials. Moreover, the equilibrium states are expanding measures. The technique consists in using an inducing scheme in a finite Markov structure with infinitely many symb...

We consider one parameter families of vector fields introduced by Rovella, obtained through modifying the eigenvalues of the geometric Lorenz attractor, replacing the expanding condition on the eigenvalues of the singularity by a contracting one. We show that there is no statistical stability within the set of parameters for which there is a physic...

We introduce random towers to study almost sure rates of correlation decay for random partially hyperbolic attractors. Using this framework, we obtain general results on almost sure exponential, stretched exponential and polynomial correlation decay rates. We then apply our results to random perturbations of systems, including Axiom A attractors, d...

We consider random perturbations of a topologically transitive local diffeomorphism of a Riemannian manifold. We show that if an absolutely continuous ergodic stationary measures is expanding (all Lyapunov exponents positive), then there is a random Gibbs-Markov-Young structure which can be used to lift that measure. We also prove that if the origi...

We consider one parameter families of vector fields introduced by Rovella, obtained through modifying the eigenvalues of the geometric Lorenz attractor, replacing the expanding condition on the eigenvalues of the singularity by a contracting one. We show that there is no statistical stability within the set of parameters for which there is a physic...

We consider some classes of piecewise expanding maps in finite dimensional spaces having invariant probability measures which are absolutely continuous with respect to Lebesgue measure. We derive an entropy formula for such measures and, using this entropy formula, in some parametrized families we present sufficient conditions for the continuity of...

We prove that for a wide family of non-uniformly hyperbolic maps and hyperbolic potentials we have equilibrium stability, i.e. the equilibrium states persist under small perturbations. For this we deduce that the topological pressure is continuous in the $C^1$ topology as a function of the dynamics and the potential. We also prove the existence of...

We prove that for a wide family of non-uniformly hyperbolic maps and hyperbolic potentials we have equilibrium stability, i.e. the equilibrium states depend continuously on the dynamics and the potential. For this we deduce that the topological pressure is continuous as a function of the dynamics and the potential. We also prove the existence of fi...

We present sufficient conditions for the (strong) statistical stability of some classes of multidimensional piecewise expanding maps. As a consequence we get that a certain natural two-dimensional extension of the classical one-dimensional family of tent maps is statistically stable.

We consider impulsive semiflows defined on compact metric spaces and deduce a variational principle. In particular, we generalize the classical notion of topological entropy to our setting of discontinuous semiflows.

We consider diffeomorphisms of compact Riemmanian manifolds which have a Gibbs-Markov-Young structure, consisting of a reference set $\Lambda$ with a hyperbolic product structure and a countable Markov partition. We assume polynomial contraction on stable leaves, polynomial backward contraction on unstable leaves, a bounded distortion property and...

We consider impulsive semiflows defined on compact metric spaces and give sufficient conditions, both on the semiflows and the potentials, for the existence and uniqueness of equilibrium states. We also generalize the classical notion of topological pressure to our setting of discontinuous semiflows and prove a variational principle.

We present an infinite dimensional Banach space in which the set of hyperbolic linear isomorphisms in that space is not dense (in the norm topology) in the set of linear isomorphisms.

We study partially hyperbolic sets $K$ on a Riemannian manifold $M$ whose tangent space splits as $T_K M=E^{cu}\oplus E^{s}$, for which the center-unstable direction $E^{cu}$ is non-uniformly expanding on some local unstable disk. We prove that the (stretched) exponential decay of recurrence times for an induced scheme can be deduced under the assu...

Lecture notes of a minicourse on SRB measures for partially hyperbolic attractors given at UFBA and UFRGS in Brazil, April 2015.

In this work we prove that hyperbolic isomorphisms in Banach spaces have a splitting into stable and unstable spaces. We also prove that hyperbolic isomorphisms are structurally stable.

We consider a diffeomorphism f of a compact manifold M which is Almost Axiom A, i.e. f is hyperbolic in a neighborhood of some compact f-invariant set, except in some singular set of neutral points. We prove that if there exists some f-invariant set of hyperbolic points with positive unstable-Lebesgue measure such that for every point in this set t...

We consider the robust family of Lorenz attractors. These attractors are chaotic in the sense that they are transitive and have sensitive dependence on the initial conditions. Moreover, they support SRB measures whose ergodic basins cover a full Lebesgue measure subset of points in the topological basin of attraction. Here we prove that the SRB mea...

We consider impulsive dynamical systems defined on compact metric spaces and their respective impulsive semiflows. We establish sufficient conditions for the existence of probability measures which are invariant by such impulsive semiflows. We also deduce the forward invariance of their non-wandering sets except the discontinuity points.

The main goal of these notes is to generalize some results which are standard for linear isomorphisms of finite dimensional vector spaces to infinite dimensional spaces. In particular, we prove that there exists an invariant splitting for any hyperbolic bounded linear isomorphism on a Banach space and that such an isomorphism is structurally stable...

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 $g:J \to J$ a...

We consider two examples of Viana maps for which the base dynamics has singularities (discontinuities or critical points) and show the existence of a unique absolutely continuous invariant probability measure and related ergodic properties such as stretched exponential decay of correlations and stretched exponential large deviations.

We present some results on the existence and continuous variation of physical measures for families of chaotic dynamical systems. Quadratic maps and Lorenz flows will be considered in more detail. A brief idea on the proof of a recent theorem in Alves and Soufi (Nonlinearity 25:3527–3552, 2012) on the statistical stability of Lorenz flows will be g...

We show that for a large class of maps on manifolds of arbitrary finite dimension, the existence of a Gibbs-Markov-Young structure (with Lebesgue as the reference measure) is a necessary as well as sufficient condition for the existence of an invariant probability measure which is absolutely continuous measure (with respect to Lebesgue) and for whi...

We consider a 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 recurrence to the critica...

A classic approach in dynamical systems is to use particular geometric structures to deduce statistical properties, for example the existence of invariant measures with stochastic-like behaviour such as large deviations or decay of correlations. Such geometric structures are generally highly non-trivial and thus a natural question is the extent to...

Let f and g be C^r unimodal maps, with r ≥ 3, topologically conjugated by h and without periodic attractors. If h is differentiable at a point p in the expanding set E(f), with h′(p)≠0, then, there is an open renormalization interval J such that h is a C r diffeomorphism in the basin B(J) of J, and h is not differentiable at any point in I ∖ B(J)....

We present conditions on families of diffeomorphisms that guarantee statistical stability and SRB entropy continuity. They rely on the existence of horseshoe-like sets with infinitely many branches and variable return times. As an application we consider the family of Henon maps within the set of Benedicks-Carleson parameters.

We consider the family of Hénon maps in the plane and show that the SRB measures vary continuously in the weak∗ topology within the set of Benedicks–Carleson parameters.

We consider a partially hyperbolic set K on a Riemannian manifold M whose tangent space splits as TKM=Ecu⊕Es, for which the center-unstable direction Ecu 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 ter...

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 convergen...

We consider smooth maps on compact Riemannian manifolds. We prove that under some mild condition of eventual volume expansion Lebesgue almost everywhere we have uniform backward volume contraction on every pre-orbit of Lebesgue almost every point. Comment: Article published in Comptes Rendu, 2006

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 attractors, for some $\alpha>0$, always have zero volume, thus extending an analogous result for uniformly hyperbolic...

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 return 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 sufficie...

We consider the family of Henon maps in the plane and show that the SRB measures vary continuously in the weak* topology within the set of Benedicks-Carleson parameters.

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 so...

The aim of this paper is to present a survey of recent results on the statistical properties of non-uniformly expanding maps on flnite-dimensional Riemannian manifolds. We will mostly focus on the existence of SRB mea- sures, continuity of the SRB measure and its entropy, decay of correlations and stochastic stability. Introduction. In broad terms,...

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 posi...

We show that there are no partially hyperbolic horseshoes with positive Lebesgue measure for diffeomorphisms whose class of differentiability is higher than 1. This generalizes a classical result by Bowen for uniformly hyperbolic horseshoes.

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 so...

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 diffeomorphi...

We show that one-dimensional maps f with strictly positive Lyapunov exponents almost everywhere admit an absolutely continuous invariant measure. If f is topologically transitive, some power of f is mixing and, in particular, the correlation of Hölder continuous observables decays to zero. The main objective of this paper is to show that the rate o...

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 co...

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. Comment: 21...

We consider smooth maps on compact Riemannian manifolds. We prove that under some mild condition of eventual volume expansion Lebesgue almost everywhere we have uniform backward volume contraction on every pre-orbit for Lebesgue almost every point.

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 co...

We consider non-uniformly expanding maps on compact Riemann- ian manifolds of arbitrary dimension, possibly having discontinuities and/or critical sets, and show that under some general conditions they admit an in- duced Markov tower structure for which the decay of the tail of the return time function can be controlled in terms of the time generic...

We consider open sets of maps in a manifold M exhibiting non-uniform expanding behaviour in some domain S 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 applicat...

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 hyperboli...

We present some recent developments on the theory of smooth dynamical systems in a Riemannian manifold exhibiting non-uniformly expanding behavior in the sense of [2]. In particular, we show that these systems have a nite number of SRB measures whose basins cover the whole manifold, and that under some uniform fast approach on the rates of expansio...

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...