arXiv subjects

Felipe García-Ramos

Publications and source records attributed to Felipe García-Ramos.

26 records · Page 2Linked to original sources

arXiv2019

Mean equicontinuity, almost automorphy and regularity

The aim of this article is to obtain a better understanding and classification of strictly ergodic topological dynamical systems with discrete spectrum. To that end, we first determine when an isomorphic maximal equicontinuous factor map of a minimal topological dynamical system has trivial (one point) fibres. In other words, we characterize when minimal mean equicontinuous systems are almost automorphic. Furthermore, we investigate another natural subclass of mean equicontinuous systems, so-called diam-mean equicontinuous systems, and show that a minimal system is diam-mean equicontinuous if and only if the maximal equicontinuous factor is regular (the points with trivial fibres have full Haar measure). Combined with previous results in the field, this provides a natural characterization for every step of a natural hierarchy for strictly ergodic topological models of ergodic systems with discrete spectrum. We also construct an example of a transitive almost diam-mean equicontinuous system with positive topological entropy, and we give a partial answer to a question of Furstenberg related to multiple recurrence.

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arXiv2019

Extender sets and measures of maximal entropy for subshifts

We prove inequalities relating the measures of maximal entropy of two patterns u,v where the extender set of u is contained in the extender set of v. Our main results are two generalizations of a Theorem of Meyerovitch; the first applies to all such v,w when G=Z, and the second to v,w with the same shape and any countable amenable finitely generated torsion-free G. As a consequence of our results we give new and simpler proofs of several facts about synchronizing subshifts and we answer a question of Climenhaga.

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arXiv2018

A hierarchy of topological systems with completely positive entropy

We define a hierarchy of systems with topological completely positive entropy in the context of continuous countable amenable group actions on compact metric spaces. For each countable ordinal we construct a dynamical system on the corresponding level of the aforementioned hierarchy and provide subshifts of finite type for the first three levels. We give necessary and sufficient conditions for entropy pairs by means of the asymptotic relation on systems with the pseudo-orbit tracing property, and thus create a bridge between a result by Pavlov and a result by Meyerovitch. As a corollary, we answer negatively an open question by Pavlov regarding necessary conditions for completely positive entropy.

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arXiv2017

When is a dynamical system mean sensitive?

This article is devoted to study which conditions imply that a topological dynamical system is mean sensitive and which do not. Among other things we show that every uniquely ergodic, mixing system with positive entropy is mean sensitive. On the other hand we provide an example of a transitive system which is cofinitely sensitive or Devaney chaotic with positive entropy but fails to be mean sensitive. As applications of our theory and examples, we negatively answer an open question regarding equicontinuity/sensitivity dichotomies raised by Tu, we introduce and present results of locally mean equicontinuous systems and we show that mean sensitivity of the induced hyperspace does not imply that of the phase space.

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arXiv2016

Mean proximality and mean Li-Yorke chaos

We prove that if a topological dynamical system is mean sensitive and contains a mean proximal pair consisting of a transitive point and a periodic point, then it is mean Li-Yorke chaotic (DC2 chaotic). On the other hand we show that a system is mean proximal if and only if it is uniquely ergodic and the unique measure is supported on one point.

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arXiv2015

Limit behaviour of $μ-$equicontinuous cellular automata

The concept of $μ-$equicontinuity was introduced by Gilman to classify cellular automata. We show that under some conditions the sequence of Cesaro averages of a measure $μ,$ converge under the actions of a $μ-$equicontinuous CA. We address questions raised by Blanchard-Tisseur on whether the limit measure is either shift-ergodic, a uniform Bernoulli measure or ergodic with respect to the CA. Many of our results hold for CA on multidimensional subshifts.

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arXiv2015

Weak forms of topological and measure theoretical equicontinuity: relationships with discrete spectrum and sequence entropy

We define weaker forms of topological and measure theoretical equicontinuity for topological dynamical systems and we study their relationships with systems with discrete spectrum and zero sequence entropy. In the topological category we show systems with zero topological sequence entropy are strictly contained in the diam-mean equicontinuous systems; and that transitive almost automorphic subshifts are diam-mean equicontinuous if and only if they are regular (i.e. the maximal equicontinuous factor map is 1-1 on a set of full Haar measure). In the measure category we show that for ergodic topological systems having discrete spectrum is equivalent of being μ-mean equicontinuous. For both categories we find characterizations using stronger versions of the classical notion of sensitivity.

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arXiv2014

A characterization of μ-equicontinuity for topological dynamical systems

Two different notions of μ-equicontinuity that apply to topological dynamical systems and probability measures were studied by Gilman (1987) and Huang-Lu-Ye (2011). One was used to classify measure preserving topological dynamical systems and the other for cellular automata. We show that if the probability space satisfies Lebesgue's density theorem and Vitali's covering theorem (for example a Cantor set or a subset of R^{d}) then both properties are equivalent. To do this we characterize when a function is Lusin measurable.

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