Invariant measures for piecewise continuous maps

Benito Pires

doi:10.1016/j.crma.2016.05.002

Dynamical systems

Invariant measures for piecewise continuous maps
[Mesures invariantes pour les applications continues par morceaux]

Présenté par : Claire Voisin

Benito Pires ¹

¹ Departamento de Computação e Matemática, Faculdade de Filosofia, Ciências e Letras, Universidade de São Paulo, 14040-901, Ribeirão Preto – SP, Brazil

Comptes Rendus. Mathématique, Volume 354 (2016) no. 7, pp. 717-722.

Résumé
Abstract

On dit que $f : [0, 1] \to [0, 1]$ est une application d'intervalle continue par morceaux s'il existe une partition $0 = x_{0} < x_{1} < \dots < x_{d} < x_{d + 1} = 1$ de $[0, 1]$ telle que $f |_{(x_{i - 1}, x_{i})}$ est continue et telle que les limites latérales $w_{0}^{+} = \lim_{x \to 0^{+}} f (x)$ , $w_{d + 1}^{-} = \lim_{x \to 1^{-}} f (x)$ , $w_{i}^{-} = \lim_{x \to x_{i}^{-}} f (x)$ et $w_{i}^{+} = \lim_{x \to x_{i}^{+}} f (x)$ existent pour chaque i. On prouve que toute application d'intervalle continue par morceaux sans connexion admet une mesure de probabilité invariante. On prouve également que toute application injective d'intervalle continue par morceaux sans connexion et sans orbite périodique est topologiquement semiconjuguée à un échange d'intervalles.

We say that $f : [0, 1] \to [0, 1]$ is a piecewise continuous interval map if there exists a partition $0 = x_{0} < x_{1} < \dots < x_{d} < x_{d + 1} = 1$ of $[0, 1]$ such that $f |_{(x_{i - 1}, x_{i})}$ is continuous and the lateral limits $w_{0}^{+} = \lim_{x \to 0^{+}} f (x)$ , $w_{d + 1}^{-} = \lim_{x \to 1^{-}} f (x)$ , $w_{i}^{-} = \lim_{x \to x_{i}^{-}} f (x)$ and $w_{i}^{+} = \lim_{x \to x_{i}^{+}} f (x)$ exist for each i. We prove that every piecewise continuous interval map without connections admits an invariant Borel probability measure. We also prove that every injective piecewise continuous interval map with no connections and no periodic orbits is topologically semiconjugate to an interval exchange transformation.

Reçu le : 2016-03-15
Accepté le : 2016-05-03
Publié le : 2016-05-24

DOI : 10.1016/j.crma.2016.05.002

Affiliations des auteurs :

Benito Pires ¹

¹ Departamento de Computação e Matemática, Faculdade de Filosofia, Ciências e Letras, Universidade de São Paulo, 14040-901, Ribeirão Preto – SP, Brazil

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     title = {Invariant measures for piecewise continuous maps},
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Benito Pires. Invariant measures for piecewise continuous maps. Comptes Rendus. Mathématique, Volume 354 (2016) no. 7, pp. 717-722. doi : 10.1016/j.crma.2016.05.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.05.002/

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José Pedro Gaivão; Michel Laurent; Arnaldo Nogueira Rotation number of 2-interval piecewise affine maps, Aequationes mathematicae (2024) | DOI:10.1007/s00010-024-01064-2
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