Invariant measures for piecewise continuous maps

Benito Pires

doi:10.1016/j.crma.2016.05.002

Dynamical systems

Invariant measures for piecewise continuous maps

Presented by: 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.

Abstract
Résumé

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.

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.

Received: 2016-03-15
Accepted: 2016-05-03
Published online: 2016-05-24

DOI: 10.1016/j.crma.2016.05.002

Author's affiliations:

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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     author = {Benito Pires},
     title = {Invariant measures for piecewise continuous maps},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {717--722},
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     volume = {354},
     number = {7},
     year = {2016},
     doi = {10.1016/j.crma.2016.05.002},
     language = {en},
}

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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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