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P. Varandas

Publications and source records attributed to P. Varandas.

5 recordsLinked to original sources
arXiv2021

Sensitivity and historic behavior for continuous maps on Baire metric spaces

We introduce a notion of sensitivity, with respect to a continuous bounded observable, which provides a sufficient condition for a continuous map, acting on a Baire metric space, to exhibit a Baire generic subset of points with historic behavior. The applications of this criterion recover, and extend, several known theorems on the genericity of the irregular set, besides yielding a number of new results, including information on the irregular set of geodesic flows, in both negative and non-positive curvature, and semigroup actions.

math.DS↗
arXiv2019

On the rotation sets of generic homeomorphisms on the torus $\mathbb T^d$

We study the rotation sets for homeomorphisms homotopic to the identity on the torus $\mathbb T^d$, $d\ge 2$. In the conservative setting, we prove that there exists a Baire residual subset of the set $\text{Homeo}_{0, λ}(\mathbb T^2)$ of conservative homeomorphisms homotopic to the identity so that the set of points with wild pointwise rotation set is a Baire residual subset in $\mathbb T^2$, and that it carries full topological pressure and full metric mean dimension. Moreover, we prove that for every $d\ge 2$ the rotation set of $C^0$-generic conservative homeomorphisms on $\mathbb T^d$ is convex. Related results are obtained in the case of dissipative homeomorphisms on tori. The previous results rely on the description of the topological complexity of the set of points with wild historic behavior and on the denseness of periodic measures for continuous maps with the gluing orbit property.

math.DS↗
arXiv2017

Stability and limit theorems for sequences of uniformly hyperbolic dynamics

In this paper we obtain an almost sure invariance principle for convergent sequences of either Anosov diffeomorphisms or expanding maps on compact Riemannian manifolds and prove an ergodic stability result for such sequences. The sequences of maps need not correspond to typical points of a random dynamical system. The methods in the proof rely on the stability of compositions of hyperbolic dynamical systems. We introduce the notion of sequential conjugacies and prove that these vary in a Lipschitz way with respect to the generating sequences of dynamical systems. As a consequence, we prove stability results for time-dependent expanding maps that complement results in [Franks74] on time-dependent Anosov diffeomorphisms.

math.DS↗
arXiv2015

Rapid mixing for the Lorenz attractor and statistical limit laws for their time-1 maps

We prove that every geometric Lorenz attractor has superpolynomial decay of correlations with respect to the unique SRB measure. Moreover, we prove the Central Limit Theorem and Almost Sure Invariance Principle for the time-1 map of the flow of geometric Lorenz attractors. In particular, our results apply to the classical Lorenz attractor.

math.DS↗