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.

Abstract (VO)
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.

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

@article{CRMATH_2016__354_7_717_0,
     author = {Benito Pires},
     title = {Invariant measures for piecewise continuous maps},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {717--722},
     publisher = {Elsevier},
     volume = {354},
     number = {7},
     year = {2016},
     doi = {10.1016/j.crma.2016.05.002},
     language = {en},
}

TY  - JOUR
AU  - Benito Pires
TI  - Invariant measures for piecewise continuous maps
JO  - Comptes Rendus. Mathématique
PY  - 2016
SP  - 717
EP  - 722
VL  - 354
IS  - 7
PB  - Elsevier
DO  - 10.1016/j.crma.2016.05.002
LA  - en
ID  - CRMATH_2016__354_7_717_0
ER  -

%0 Journal Article
%A Benito Pires
%T Invariant measures for piecewise continuous maps
%J Comptes Rendus. Mathématique
%D 2016
%P 717-722
%V 354
%N 7
%I Elsevier
%R 10.1016/j.crma.2016.05.002
%G en
%F CRMATH_2016__354_7_717_0

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/

Bibliographie
Cité par

[1] R. Adler; L. Flatto Geodesic flows, interval maps, and symbolic dynamics, Bull. Amer. Math. Soc., Volume 25 (1991), pp. 229-334

[2] L. Alsedà; M. Misiurewicz Semiconjugacy to a map of constant slope, Discrete Contin. Dyn. Syst., Ser. B, Volume 20 (2015), pp. 3403-3413

[3] V. Baladi Positive Transfer Operators and Decay of Correlations, Advanced Series in Nonlinear Dynamics, World Scientific Publishing Company, 2000

[4] R. Bowen Invariant measures for Markov maps of the interval, Commun. Math. Phys., Volume 69 (1979), pp. 1-17

[5] A. Boyarsky; P. Góra Laws of Chaos: Invariant Measures and Dynamical Systems in One Dimension, Birkhäuser, 1997

[6] C. Gutierrez Smoothing continuous flows on two-manifolds and recurrences, Ergod. Theory Dyn. Syst., Volume 6 (1986) no. 1, pp. 17-44

[7] M. Keane Interval exchange transformations, Math. Z., Volume 141 (1975) no. 1, pp. 25-31

[8] N. Kryloff; N. Bogoliouboff La théorie générale de la mesure dans son application à l'étude des systèmes dynamiques de la mécanique non linéaire, Ann. of Math. (2), Volume 38 (1937) no. 1, pp. 65-113

[9] A. Lasota; J.A. Yorke On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc., Volume 186 (1973), pp. 481-488

[10] C. Liverani Invariant measures and their properties. A functional analytic point of view, Dynamical Systems. Part II, Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Norm. Sup., Pisa, 2003, pp. 185-237

[11] S. Marmi; P. Moussa; J-C. Yoccoz Linearization of generalized interval exchange maps, Ann. of Math. (2), Volume 176 (2012), pp. 1583-1646

[12] J. Milnor; W. Thurston On iterated maps of the interval, Lecture Notes in Mathematics, vol. 1342, Springer, Berlin, 1988, pp. 465-563

[13] M. Misiurewicz; S. Roth No semiconjugacy to a map of constant slope, Ergod. Theory Dyn. Syst., Volume 36 (2016), pp. 875-889

[14] A. Nogueira; B. Pires Dynamics of piecewise contractions of the interval, Ergod. Theory Dyn. Syst., Volume 35 (2015), pp. 2198-2215

[15] K. Parthasarathy Probability Measures on Metric Spaces, American Mathematical Society, 2005

[16] P. Walters An Introduction to Ergodic Theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, 2000

Cité par Sources :

Commentaires - Politique