arXiv subjects

Gabriel Lacerda

Publications and source records attributed to Gabriel Lacerda.

3 recordsLinked to original sources

arXiv2026

Generalized entropy of measure-induced maps

A classical result by E. Glasner and B. Weiss states that the topological entropy of a map $f$ is zero if and only if the topological entropy of its measure-induced map $f_*$ is zero, where $f_*$ is defined as the push-forward of a measure. In this work, we use generalized entropy to distinguish the complexity of these maps and prove that the measure-induced map is much more complex than the original map. Moreover, we introduce the generalized mean dimension, an invariant that is useful for distinguishing dynamical systems with zero mean dimension, including those with the small-boundary property, and we show a relationship between this new invariant and generalized entropy.

math.DS↗

arXiv2025

Mean dimension explosion of induced homeomorphisms

Given $X$ a compact metric space and $T: X \to X$ a continuous map, the induced hyperspace map $T_\mathcal{K}$ acts on the hyperspace $\mathcal{K}(X)$ of closed and nonempty subsets of $X$, and on the continuum hyperspace $\mathcal{C}(X) \subset \mathcal{K}(X)$ of connected sets. This work studies the mean dimension explosion phenomenon: when the base system $T$ has zero topological entropy, but the mean dimension of the induced map $T_\mathcal{K}$ is infinite. In particular, this phenomenon occurs for Morse-Smale diffeomorphisms. Furthermore, for a circle homeomorphism $H$, the mean dimension explosion does not occur if and only if $H$ is conjugate to a rotation. For the metric mean dimension, a different result is obtained: we establish sufficient conditions for the induced hyperspace map to have zero or infinite metric mean dimension.

math.DS↗

arXiv2023

Typical conservative homeomorphisms have total metric mean dimension

Given a compact smooth boundaryless manifold with dimension greater than one endowed with a locally positive non-atomic measure $μ$, we prove that typical $μ$-preserving homeomorphisms have upper metric mean dimension, with respect to the Riemannian distance, equal to the dimension of the manifold. Moreover, we prove that $μ$ is a measure of maximal metric mean dimension, with respect to the variational principle established in [VV17].

math.DS↗