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Vilton Pinheiro

Publications and source records attributed to Vilton Pinheiro.

At least 19 recordsLinked to original sources
arXiv2025

Multidimensional non-uniform hyperbolicity, robust exponential mixing and the basin problem

We show that the ergodic, topological and geometric basins coincide for hyperbolic dominated ergodic $cu$-Gibbs states, solving the ``basin problem'' for a wide class of non-uniformly hyperbolic systems. We obtain robust examples of exponential mixing physical measures for systems with multidimensional nonuniform hyperbolic dominated splitting, without uniformly expanding or contracting subbundles. Both results are a consequence of extending the construction of Gibbs-Markov-Young structures from partial hyperbolic systems to systems with only a dominated splitting, using the existence of an ``improved hyperbolic block'', with respect to Pesin's Nonuniform Hyperbolic Theory, for hyperbolic dominated measures of smooth maps, obtained through hyperbolic times and associated ``coherent schedules'' introduced by one of the coauthors.

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arXiv2022

Thermodynamic formalism for expanding measures

In this paper we study the thermodynamic formalism of strongly transitive endomorphisms $f$, focusing on the set all expanding measures. In case $f$ is a non-flat $C^{1+}$ map defined on a Riemannian manifold, these are invariant probability measures with all its Lyapunov exponents positive. Given a H\"older continuous potential $\varphi$ we prove the uniqueness of the equilibrium state among the space of expanding measures. Moreover, we show that the existence of an expanding measure $\mu$ maximizing the entropy on the the space of expanding measures implies the existence and uniqueness of equilibrium state $\mu_{\varphi}$ on the space of expanding measures for any H\"older continuous potential $\varphi$ with a small oscillation $\text{osc }\varphi=\sup\varphi-\inf\varphi$. As some applications, we prove that Collet-Eckmann quadratic maps does not admit phase transition for H\"older potential, and show that for Viana maps and every H\"older continuous potential of sufficiently small oscillation has a unique equilibrium state.

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arXiv2022

Super-expanding measures

We study the one-dimensional expanding Lorenz maps and show the existence of dense subset D of Lorens maps such that each f in D has an uncountable set of ergodic invariant probabilities with infinite Lyapunov exponent and positive entropy. Such measures may appear when the singularity has fast recurrence to itself. Conversely, if the singularity has slow recurrence to itself then the Lorenz map has an upper bound to the Lyapunov exponent of all invariant measures.

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arXiv2021

Topological and statistical attractors for interval maps

We use the concept of Baire Ergodicity and Ergodic Formalism introduced to study topological and statistical attractors for interval maps, even with discontinuities. For that we also analyze the {\em wandering intervals attractors}. As a result, we establish the finiteness of the non-periodic topological attractors for piecewise $C^2$ maps with discontinuities. For $C^2$ interval maps without discontinuities, we show the coincidence of the statistical attractors with the topological ones and we calculate the upper Birkhoff averages of continuous functions for generic points, even when the map has abundance of historical behavior.

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arXiv2021

Ergodic Formalism for topological Attractors and historic behavior

We introduce the concepts of Baire Ergodicity and Ergodic Formalism, employing them to study topological and statistical attractors. Specifically, we establish the existence and finiteness of such attractors and provide applications for maps of the interval, Viana maps, non-uniformly expanding maps, partially hyperbolic systems, strongly transitive dynamics, and skew-products. In a dynamical system with an abundance of historic behavior (encompassing all systems with some hyperbolicity, particularly Axiom A systems), one can show the existence of a residual set with zero measure for every invariant probability measure. Hence, in principle, utilizing the classical ergodic theory to control the asymptotic topological/statistical behavior of generic orbits is not feasible. Nevertheless, the results presented here can also be applied to such a system, contributing to the study of generic orbits in systems with an abundance of historic behavior.

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arXiv2018

Lift and Synchronization

We study the problem of lifting a measure to an induced map $F(x)=f^{R(x)}(x)$. In particular, we give a necessary and sufficient condition for an ergodic $f$ invariant probability $\mu$ to be $F$-liftable as well as a condition for the lift to be an ergodic measure. Moreover, we show that every lift of $\mu$ is a weighted average of the restriction of $\mu$ to a countable number of $F$-ergodic components. We introduce the concept of a coherent schedule of events and relate it to the lift problem. As a consequence, we prove that we can always synchronize coherent schedules at almost every point with respect to a given invariant probability $\mu$, showing that we can synchronize `Pliss times' $\mu$ almost everywhere. We also provide a version of this synchronization to non-invariant measures and, from that, we obtain some results related to Viana's conjecture on the existence of SRB measures for maps with non-zero Lyapunov exponents for Lebesgue almost every point.

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arXiv2016

Abundance of wild historic behavior

Using Caratheodory measures, we associate to each positive orbit ${\mathcal O}_{f}^{+}(x)$ of a measurable map $f$, a Borel measure $\eta_{x}$. We show that $\eta_{x}$ is $f$-invariant whenever $f$ is continuous or $\eta_{x}$ is a probability. These measures are used to study the \emph{historic} points of the system, that is, \emph{points with no Birkhoff averages}, and we construct topologically generic subset of \emph{wild historic points} for wide classes of dynamical models. We use properties of the measure $\eta_x$ to deduce some features of the dynamical system involved, like the \emph{existence of heteroclinic connections from the existence of open sets of historic points}.

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arXiv2016

On the Finiteness of Attractors for piecewise $C^2$ Maps of the Interval

We consider piecewise $C^2$ non-flat maps of the interval and show that, for Lebesgue almost every point, its omega-limit set is either a periodic orbit, a cycle of intervals or the closure of the orbits of a subset of the critical points. In particular, every piecewise $C^2$ non-flat map of the interval displays only a finite number of non-periodic attractors.

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arXiv2016

On the Finiteness of Attractors for One-Dimensional Maps with Discontinuities

Since the proof, at the end of the 80's, of the finiteness of the number of attractors for $C^3$ maps of the interval having negative Schwarzian derivative, it has been generally considered that the same result could be true for maps with discontinuities. In the present paper we show that this is indeed the case.

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arXiv2014

Contribution to the ergodic theory of robustly transitive maps

In this article we intend to contribute in the understanding of the ergodic properties of the set RT of robustly transitive local diffeomorphisms on a compact manifold M without boundary. We prove that there exists a C^1 residual subset R_0 of RT such that any f in R_0 has a residual subset of M with dense pre-orbits. Moreover, C^1 generically in the space of local diffeomorphisms with no splitting and all points with dense pre-orbit, there are uncountably many ergodic expanding invariant measures with full support and exhibiting exponential decay of correlations. In particular, these results hold for an important class of robustly transitive maps considered in [Lizana-Pujals'12].

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arXiv2012

Thermodynamic formalism for Lorenz maps

For a 2-dimensional map representing an expanding geometric Lorenz at- tractor we prove that the attractor is the closure of a union of as long as possible unstable leaves with ending points. This allows to define the notion of good measures, those giving full measure to the union of these open leaves. Then, for any Hölder continuous potential we prove that there exists at most one relative equilibrium state among the set of good measures. Condition yielding existence are given.

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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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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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arXiv2008

Expanding Measures

We prove that any C^{1+} transformation, possibly with a (non-flat) critical or singular region, admits an invariant probability measure absolutely continuous with respect to any expanding measure whose Jacobian satisfies a mild distortion condition. This is an extension to arbitrary dimension of a famous theorem of Keller for maps of the interval with negative Schwarzian derivative. We also show how to construct an induced Markov map F such that every expanding probability of the initial transformation lifts to an invariant probability of F. The induced time is bounded at each point by the corresponding first hyperbolic time (the first time the dynamics exhibits hyperbolic behavior). In particular, F may be used to study decay of correlations and others statistical properties of the initial map, relative to any expanding probability.

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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

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 done by A. Katok, in a related context.

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arXiv2006

SRB measures for weakly expanding maps

We construct SRB measures for endomorphisms satisfying conditions far weaker than the non-uniformly expansion. As a consequence, the definition of non-uniformly expanding map can be weakened. We also prove the existence of an absolutely continuous invariant measure for local diffeomorphisms, only assuming the existence of hyperbolic times for Lebesgue almost all point of the manifold.

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