arXiv subjects

Rômulo M. Vermersch

Publications and source records attributed to Rômulo M. Vermersch.

4 recordsLinked to original sources

arXiv2026

On the entropy of processes generated by quasifactors

Let $(X,\mathcal{X},μ,T)$ be a measurable dynamical system, where $T$ is a homeomorphism acting on the zero-dimensional compact metric space $X$, $\mathcal{X}$ is the Borel $σ$-algebra on $X$ and $μ$ is an ergodic $T$-invariant Borel probability measure on $X$. In this work we show that, if $h_μ(T)>0$, then $h_{\widetildeμ}(\widetilde{T})>0$ for every ergodic quasifactor $\widetildeμ$ of $μ$ having full-support. This result can be seen, in the zero-dimensional setting, as a counterpart of the Glasner-Weiss result about quasifactors of zero-entropy systems.

math.DS↗

arXiv2023

Measure-theoretic Uniformly Positive Entropy on the Space of Probability Measures

For a homeomorphism $T$ on a compact metric space $X$, a $T$-invariant Borel probability measure $μ$ on $X$ and a measure-theoretic quasifactor $\widetildeμ$ of $μ$, we study the relationship between the local entropy of the system $(X,μ,T)$ and of its induced system $(\mathcal{M}(X),\widetildeμ,\widetilde{T})$, where $\widetilde{T}$ is the homeomorphism induced by $T$ on the space $\mathcal{M}(X)$ of all Borel probability measures defined on $X$.

math.DS↗

arXiv2020

Uniformly Positive Entropy of Induced Transformations

Let $(X,T)$ be a topological dynamical system consisting of a compact metric space $X$ and a continuous surjective map $T : X \to X$. By using local entropy theory, we prove that $(X,T)$ has uniformly positive entropy if and only if so does the induced system $(\cM(X),\wt{T})$ on the space of Borel probability measures endowed with the weak$^*$ topology. This result can be seen as a version for the notion of uniformly positive entropy of the corresponding result for topological entropy due to Glasner and Weiss.

math.DS↗

arXiv2014

On the Dynamics of Induced Maps on the Space of Probability Measures

For the generic continuous map and for the generic homeomorphism of the Cantor space, we study the dynamics of the induced map on the space of probability measures, with emphasis on the notions of Li-Yorke chaos, topological entropy, equicontinuity, chain continuity, chain mixing, shadowing and recurrence. We also establish some results concerning induced maps that hold on arbitrary compact metric spaces.

math.DS↗