Cortez, María Isabel; Rivera-Letelier, Juan Choquet simplices as spaces of invariant probability measures on post-critical sets. (English) Zbl 1192.37053 Ann. Inst. Henri Poincaré, Anal. Non Linéaire 27, No. 1, 95-115 (2010). The paper under review deals with topological dynamical systems. In this setting, it is well known that the space of invariant probability measures (i.p.m.’s) is a non-empty metrizable Choquet simplex (respect to the weak* topology). Reciprocally, T. Downarowicz [Isr. J. Math. 74, No. 2–3, 241–256 (1991; Zbl 0746.58047)] proved that every non-empty metrizable Choquet simplex \(\mathcal{C}\) can be realized (up to an affine homeomorphism) as the space of i.p.m.’s of the dynamical system generated by a minimal homeomorphism of a Cantor set.Now, the authors give a more natural realization of \(\mathcal{C},\) since they prove that it arises as the space of i.p.m.’s associated to some \(\omega\)-limit sets of logistic maps. Namely, if \(f_{\lambda}(x)=\lambda x(1-x),\) \(x\in [0,1],\) and \(X_\lambda\) is the \(\omega\)-limit set of \(x=\frac{1}{2}\) respect to \(f_\lambda,\) it holds:Main Theorem. For each \(\mathcal{C}\) there is a parameter \(\lambda\in (0,4]\) such that the post-critical set \(X_\lambda\) is a Cantor set, the restriction of \(f_\lambda\) to \(X_\lambda\) is minimal, and such that the space of i.p.m.’s supported by this set, endowed with the weak* topology, is affine homeomorphic to \(\mathcal{C}.\) The proof of this result relies on the notions of cutting time sequences and kneading maps [see H. Bruin, G. Keller and M. St. Pierre, Ergodic Theory Dyn. Syst. 17, No. 6, 1267–1287 (1997; Zbl 0898.58012)]. Reviewer: Antonio Linero Bas (Murcia) Cited in 5 Documents MSC: 37E05 Dynamical systems involving maps of the interval 37A99 Ergodic theory 37B10 Symbolic dynamics 54H20 Topological dynamics (MSC2010) 54E99 Topological spaces with richer structures Keywords:logistic map; post-critical set; invariant measures; ergodic measures; Choquet simplices; weak* topology; minimal Cantor system; generalized odometer; Toeplitz flows; cutting times; kneading map Citations:Zbl 0746.58047; Zbl 0898.58012 PDFBibTeX XMLCite \textit{M. I. Cortez} and \textit{J. Rivera-Letelier}, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 27, No. 1, 95--115 (2010; Zbl 1192.37053) Full Text: DOI arXiv References: [1] Alfsen, Erik M., Compact Convex Sets and Boundary Integrals, Ergeb. Math. Grenzgeb., vol. 57 (1971), Springer-Verlag: Springer-Verlag New York · Zbl 0209.42601 [2] Barat, Guy; Downarowicz, Tomasz; Liardet, Pierre, Dynamiques associées à une échelle de numération, Acta Arith., 103, 1, 41-78 (2002) · Zbl 1002.37005 [3] Block, Louis; Keesling, James; Misiurewicz, Michał, Strange adding machines, Ergodic Theory Dynam. Systems, 26, 3, 673-682 (2006) · Zbl 1096.37005 [4] Bruin, Henk; Keller, Gerhard; St. Pierre, Matthias, Adding machines and wild attractors, Ergodic Theory Dynam. Systems, 17, 6, 1267-1287 (1997) · Zbl 0898.58012 [5] Blokh, A. M.; Lyubich, M. Yu., Measurable dynamics of \(S\)-unimodal maps of the interval, Ann. Sci. École Norm. Sup. (4), 24, 5, 545-573 (1991) · Zbl 0790.58024 [6] Bruin, H., Combinatorics of the kneading map, (Thirty Years After Sharkovskiĭ’s Theorem: New Perspectives. Thirty Years After Sharkovskiĭ’s Theorem: New Perspectives, Murcia, 1994. Thirty Years After Sharkovskiĭ’s Theorem: New Perspectives. Thirty Years After Sharkovskiĭ’s Theorem: New Perspectives, Murcia, 1994, World Sci. Ser. Nonlinear Sci. Ser. B Spec. Theme Issues Proc., vol. 8 (1995), World Sci. Publ.: World Sci. Publ. River Edge, NJ), 77-87, reprint of the paper reviewed in MR1361922 (96k:58070) · Zbl 0886.58023 [7] Bruin, H., Minimal Cantor systems and unimodal maps, J. Difference Equ. Appl., 9, 3-4, 305-318 (2003), dedicated to Professor Alexander N. Sharkovsky on the occasion of his 65th birthday · Zbl 1026.37003 [8] Cortez, María Isabel, Realization of a Choquet simplex as the set of invariant probability measures of a tiling system, Ergodic Theory Dynam. Systems, 26, 5, 1417-1441 (2006) · 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; 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] Durand, F.; Host, B.; Skau, C., Substitutional dynamical systems, Bratteli diagrams and dimension groups, Ergodic Theory Dynam. Systems, 19, 4, 953-993 (1999) · Zbl 1044.46543 [11] Downarowicz, Tomasz, The Choquet simplex of invariant measures for minimal flows, Israel J. Math., 74, 2-3, 241-256 (1991) · Zbl 0746.58047 [12] Downarowicz, Tomasz, Survey of odometers and Toeplitz flows, (Algebraic and Topological Dynamics. Algebraic and Topological Dynamics, Contemp. Math., vol. 385 (2005), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 7-37 · Zbl 1096.37002 [13] Effros, Edward G., Dimensions and \(C^*\)-Algebras, CBMS Reg. Conf. Ser. Math., vol. 46 (1981), Conference Board of the Mathematical Sciences: Conference Board of the Mathematical Sciences Washington, DC · Zbl 0475.46050 [14] Gjerde, Richard; Johansen, Ørjan, Bratteli-Vershik models for Cantor minimal systems: Applications to Toeplitz flows, Ergodic Theory Dynam. Systems, 20, 6, 1687-1710 (2000) · Zbl 0992.37008 [15] Glasner, Eli, Ergodic Theory via Joinings, Math. Surveys Monogr., vol. 101 (2003), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 1038.37002 [16] Grabner, Peter J.; Liardet, Pierre; Tichy, Robert F., Odometers and systems of numeration, Acta Arith., 70, 2, 103-123 (1995) · Zbl 0822.11008 [17] Gambaudo, Jean-Marc; Martens, Marco, Algebraic topology for minimal Cantor sets, Ann. Henri Poincaré, 7, 3, 423-446 (2006) · Zbl 1090.37006 [18] Giordano, Thierry; Putnam, Ian F.; Skau, Christian F., Topological orbit equivalence and \(C^\ast \)-crossed products, J. Reine Angew. Math., 469, 51-111 (1995) · Zbl 0834.46053 [19] Haydon, Richard, A new proof that every Polish space is the extreme boundary of a simplex, Bull. London Math. Soc., 7, 97-100 (1975) · Zbl 0302.46003 [20] Hofbauer, Franz, The topological entropy of the transformation \(x \mapsto a x(1 - x)\), Monatsh. Math., 90, 2, 117-141 (1980) · Zbl 0433.54009 [21] Herman, Richard H.; Putnam, Ian F.; Skau, Christian F., Ordered Bratteli diagrams, dimension groups and topological dynamics, Internat. J. Math., 3, 6, 827-864 (1992) · Zbl 0786.46053 [22] Ledrappier, François, Some properties of absolutely continuous invariant measures on an interval, Ergodic Theory Dynam. Systems, 1, 1, 77-93 (1981) · Zbl 0487.28015 [23] Ledrappier, François, Quelques propriétés ergodiques des applications rationnelles, C. R. Acad. Sci. Paris Sér. I Math., 299, 1, 37-40 (1984) · Zbl 0567.58016 [24] Lazar, A. J.; Lindenstrauss, J., Banach spaces whose duals are \(L_1\) spaces and their representing matrices, Acta Math., 126, 165-193 (1971) · Zbl 0209.43201 [25] Mañé, Ricardo, The Hausdorff dimension of invariant probabilities of rational maps, (Dynamical Systems. Dynamical Systems, Valparaiso, 1986. Dynamical Systems. Dynamical Systems, Valparaiso, 1986, Lecture Notes in Math., vol. 1331 (1988), Springer: Springer Berlin), 86-117 · Zbl 0658.58015 [26] Ormes, Nicholas S., Strong orbit realization for minimal homeomorphisms, J. Anal. Math., 71, 103-133 (1997) · Zbl 0881.28013 [27] Przytycki, Feliks, Lyapunov characteristic exponents are nonnegative, Proc. Amer. Math. Soc., 119, 1, 309-317 (1993) · Zbl 0787.58037 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.