Choquet simplices as spaces of invariant probability measures on post-critical sets

Cortez, María Isabel ; Rivera-Letelier, Juan

Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010), p. 95-115 / Harvested from Numdam

Résumé

Une conséquence bien connue du théorème de décomposition ergodique est que l'espace des mesures de probabilité invariantes d'un système dynamique topologique est un simplexe de Choquet métrisable et non vide. On montre que tout simplexe de Choquet métrisable et non vide se réalise comme l'espace des mesures de probabilité invariantes sur l'ensemble post-critique d'une application logistique. Ici, l'ensemble post-critique d'une application logistique est l'ensemble ω-limite de son unique point critique. En effet, on démontre que l'application logistique f peut être choisie de telle façon que son ensemble post-critique soit un ensemble de Cantor où f est minimal, et tel que chaque mesure de probabilité invariante sur cet ensemble soit d'exposant de Lyapunov null, et un état d'équilibre pour le potentiel $ -\mathrm{ln}|{f}^{\prime }|$.

Abstract

A well-known consequence of the ergodic decomposition theorem is that the space of invariant probability measures of a topological dynamical system, endowed with the weak∗ topology, is a non-empty metrizable Choquet simplex. We show that every non-empty metrizable Choquet simplex arises as the space of invariant probability measures on the post-critical set of a logistic map. Here, the post-critical set of a logistic map is the ω-limit set of its unique critical point. In fact we show the logistic map f can be taken in such a way that its post-critical set is a Cantor set where f is minimal, and such that each invariant probability measure on this set has zero Lyapunov exponent, and is an equilibrium state for the potential $ -\mathrm{ln}|{f}^{\prime }|$.

Détail

Publié le : 2010-01-01

Zbl 1192.37053

DOI : https://doi.org/10.1016/j.anihpc.2009.07.008
Classification:  37E05,  37A99,  37B10,  54H20

Bibliographie

[1] Erik M. Alfsen, Compact Convex Sets and Boundary Integrals, Ergeb. Math. Grenzgeb. vol. 57, Springer-Verlag, New York (1971) | Zbl 0209.42601

[2] Guy Barat, Tomasz Downarowicz, Pierre Liardet, Dynamiques associées à une échelle de numération, Acta Arith. 103 no. 1 (2002), 41-78 | Zbl 1002.37005

[3] Louis Block, James Keesling, Michał Misiurewicz, Strange adding machines, Ergodic Theory Dynam. Systems 26 no. 3 (2006), 673-682 | Zbl 1096.37005

[4] Henk Bruin, Gerhard Keller, Matthias St. Pierre, Adding machines and wild attractors, Ergodic Theory Dynam. Systems 17 no. 6 (1997), 1267-1287 | Zbl 0898.58012

[5] A.M. Blokh, M.Yu. Lyubich, Measurable dynamics of S-unimodal maps of the interval, Ann. Sci. École Norm. Sup. (4) 24 no. 5 (1991), 545-573 | Numdam | Zbl 0790.58024

[6] H. Bruin, Combinatorics of the kneading map, Thirty Years After Sharkovskiĭ's Theorem: New Perspectives, Murcia, 1994, World Sci. Ser. Nonlinear Sci. Ser. B Spec. Theme Issues Proc. vol. 8, World Sci. Publ., River Edge, NJ (1995), 77-87 | Zbl 0886.58023

[7] H. Bruin, Minimal Cantor systems and unimodal maps, J. Difference Equ. Appl. 9 no. 3–4 (2003), 305-318 | Zbl 1026.37003

[8] María Isabel Cortez, Realization of a Choquet simplex as the set of invariant probability measures of a tiling system, Ergodic Theory Dynam. Systems 26 no. 5 (2006), 1417-1441 | Zbl 1138.37009

[9] María Isabel Cortez, Juan Rivera-Letelier, Invariant measures of minimal post-critical sets of logistic maps, Israel. J. Math., in press, arXiv:0804.4550v1 | Zbl 1202.37021

[10] F. Durand, B. Host, C. Skau, Substitutional dynamical systems, Bratteli diagrams and dimension groups, Ergodic Theory Dynam. Systems 19 no. 4 (1999), 953-993 | Zbl 1044.46543

[11] Tomasz Downarowicz, The Choquet simplex of invariant measures for minimal flows, Israel J. Math. 74 no. 2–3 (1991), 241-256 | Zbl 0746.58047

[12] Tomasz Downarowicz, Survey of odometers and Toeplitz flows, Algebraic and Topological Dynamics, Contemp. Math. vol. 385, Amer. Math. Soc., Providence, RI (2005), 7-37 | Zbl 1096.37002

[13] Edward G. Effros, Dimensions and $ {C}^{*}$-Algebras, CBMS Reg. Conf. Ser. Math. vol. 46, Conference Board of the Mathematical Sciences, Washington, DC (1981) | Zbl 0475.46050

[14] Richard Gjerde, Ørjan Johansen, Bratteli–Vershik models for Cantor minimal systems: Applications to Toeplitz flows, Ergodic Theory Dynam. Systems 20 no. 6 (2000), 1687-1710 | Zbl 0992.37008

[15] Eli Glasner, Ergodic Theory via Joinings, Math. Surveys Monogr. vol. 101, Amer. Math. Soc., Providence, RI (2003) | Zbl 1038.37002

[16] Peter J. Grabner, Pierre Liardet, Robert F. Tichy, Odometers and systems of numeration, Acta Arith. 70 no. 2 (1995), 103-123 | Zbl 0822.11008

[17] Jean-Marc Gambaudo, Marco Martens, Algebraic topology for minimal Cantor sets, Ann. Henri Poincaré 7 no. 3 (2006), 423-446 | Zbl 1090.37006

[18] Thierry Giordano, Ian F. Putnam, Christian F. Skau, Topological orbit equivalence and $ {C}^{\ast }$-crossed products, J. Reine Angew. Math. 469 (1995), 51-111 | Zbl 0834.46053

[19] Richard Haydon, A new proof that every Polish space is the extreme boundary of a simplex, Bull. London Math. Soc. 7 (1975), 97-100 | Zbl 0302.46003

[20] Franz Hofbauer, The topological entropy of the transformation $ x\mapsto ax(1-x)$, Monatsh. Math. 90 no. 2 (1980), 117-141 | Zbl 0433.54009

[21] Richard H. Herman, Ian F. Putnam, Christian F. Skau, Ordered Bratteli diagrams, dimension groups and topological dynamics, Internat. J. Math. 3 no. 6 (1992), 827-864 | Zbl 0786.46053

[22] François Ledrappier, Some properties of absolutely continuous invariant measures on an interval, Ergodic Theory Dynam. Systems 1 no. 1 (1981), 77-93 | Zbl 0487.28015

[23] François Ledrappier, Quelques propriétés ergodiques des applications rationnelles, C. R. Acad. Sci. Paris Sér. I Math. 299 no. 1 (1984), 37-40 | Zbl 0567.58016

[24] A.J. Lazar, J. Lindenstrauss, Banach spaces whose duals are $ {L}_{1}$ spaces and their representing matrices, Acta Math. 126 (1971), 165-193 | Zbl 0209.43201

[25] Ricardo Mañé, The Hausdorff dimension of invariant probabilities of rational maps, Dynamical Systems, Valparaiso, 1986, Lecture Notes in Math. vol. 1331, Springer, Berlin (1988), 86-117 | Zbl 0658.58015

[26] Nicholas S. Ormes, Strong orbit realization for minimal homeomorphisms, J. Anal. Math. 71 (1997), 103-133 | Zbl 0881.28013

[27] Feliks Przytycki, Lyapunov characteristic exponents are nonnegative, Proc. Amer. Math. Soc. 119 no. 1 (1993), 309-317 | Zbl 0787.58037