Invariant measures for piecewise continuous maps

Pires, Benito

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

Pires, Benito 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, Tome 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 :

Pires, Benito 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 = {Pires, Benito},
     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},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2016.05.002/}
}

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

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

Pires, Benito. Invariant measures for piecewise continuous maps. Comptes Rendus. Mathématique, Tome 354 (2016) no. 7, pp. 717-722. doi : 10.1016/j.crma.2016.05.002. http://www.numdam.org/articles/10.1016/j.crma.2016.05.002/

Bibliographie
Cité par

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Cité par Sources :