arXiv subjects

Alexandre Trilles

Publications and source records attributed to Alexandre Trilles.

4 recordsLinked to original sources

arXiv2025

On the Invariance of Expansive Measures for Flows

We study expansive measures for continuous flows without fixed points on compact metric spaces. We provide a new characterization of expansive measures through dynamical balls that, in contrast to the dynamical balls considered in [\emph{J. Differ. Equ.}, 256 (2014):2246--2260], are actually Borel sets. This makes the theory more amenable to measure-theoretic analysis. We prove that every ergodic invariant measure with positive entropy is positively expansive, extending the results of \emph{Ergod. Th. \& Dynam. Sys.} \textbf{4}(3) (2014):765--776] to the setting of flows. This implies that flows with positive topological entropy admit expansive invariant measures. Furthermore, we show that the stable classes of such measures have zero measure. Lastly, we prove that the set of expansive measures for a flow is a $G_{\delta\sigma}$-subset of the space of all probability measures and that every expansive measure (invariant or not) can be approximated by expansive measures supported on invariant sets.

math.DS↗

arXiv2024

On the weakness of the vague specification property

We show that the vague specification property is strictly weaker than most of the specification-like properties, by establishing its equivalence with the asymptotic average shadowing property. In particular, we see that the weak specification property implies the vague specification property, but the converse does not hold, answering the question posed by Downarowicz and Weiss in [Ergod. Th. \& Dynam. Sys. 44(9) (2024), 2565--2580]. Additionally, we prove that, for surjective systems, the asymptotic average shadowing property is equivalent to the average shadowing property if the phase space is complete with respect to the dynamical Besicovitch pseudometric. We use the combination of both results to prove that the proximal and minimal shift spaces from [Ergod. Th. \& Dynam. Sys., 45(2) (2025), 396--426] possess the vague specification property (asymptotic average shadowing property). Our findings also allow us to address a couple of questions from [Fund. Math., 224(3) (2014), 241--278] about the asymptotic average shadowing property.

math.DS↗

arXiv2024

A characterization of zero entropy loosely Bernoulli flows via FK-pseudometric

We introduce the Feldman-Katok pseudometric (FK-pseudometric for short) for flows. We then provide a characterization of zero entropy loosely Bernoulli measures for continuous flows via the FK-pseudometric extending the result known for discrete-time dynamical systems. We also provide a purely topological characterization of uniquely ergodic continuous flows whose unique invariant measure is zero entropy loosely Bernoulli.

math.DS↗

arXiv2020

On the Space of Iterated Function Systems and Their Topological Stability

We study iterated function systems (IFS) with compact parameter space. We show that the space of IFS with phase space $X$ is the hyperspace of the space of self continuous maps of $X$. With this result we obtain that the Hausdorff distance is a natural metric for this space which we use to define topological stability. Then we prove, in the context of IFS, the classical results showing that shadowing property is a necessary condition for topological stability and shadowing property added to expansiveness are a sufficient condition for topological stability. To prove these statements, in fact, we use a stronger type of shadowing, called concordant shadowing property. We also give an example showing that concordant shadowing property is truly different than the traditional definition of shadowing property for IFS.

math.DS↗