Number of equilibrium states of piecewise monotonic maps of the interval
HTML articles powered by AMS MathViewer
- by Jérôme Buzzi PDF
- Proc. Amer. Math. Soc. 123 (1995), 2901-2907 Request permission
Erratum: Proc. Amer. Math. Soc. 125 (1997), 3131-3131.
Abstract:
We prove a bound of the form suggested by S. Newhouse for the number of measures with maximal entropy for a piecewise monotonic map with N monotonicity intervals: $4(N - 1)$. More generally we consider a potential $\phi$ of bounded distortion. If $\sup \phi < P(f,\phi )$, we give an explicit bound in terms of N and of the pressure.References
- Manfred Denker, Christian Grillenberger, and Karl Sigmund, Ergodic theory on compact spaces, Lecture Notes in Mathematics, Vol. 527, Springer-Verlag, Berlin-New York, 1976. MR 0457675, DOI 10.1007/BFb0082364
- Manfred Denker and Mariusz Urbański, On the existence of conformal measures, Trans. Amer. Math. Soc. 328 (1991), no. 2, 563–587. MR 1014246, DOI 10.1090/S0002-9947-1991-1014246-4
- Manfred Denker, Gerhard Keller, and Mariusz Urbański, On the uniqueness of equilibrium states for piecewise monotone mappings, Studia Math. 97 (1990), no. 1, 27–36. MR 1074766, DOI 10.4064/sm-97-1-27-36
- Franz Hofbauer, On intrinsic ergodicity of piecewise monotonic transformations with positive entropy, Israel J. Math. 34 (1979), no. 3, 213–237 (1980). MR 570882, DOI 10.1007/BF02760884
- Franz Hofbauer, Kneading invariants and Markov diagrams, Ergodic theory and related topics (Vitte, 1981) Math. Res., vol. 12, Akademie-Verlag, Berlin, 1982, pp. 85–95. MR 730773
- Gerhard Keller, Lifting measures to Markov extensions, Monatsh. Math. 108 (1989), no. 2-3, 183–200. MR 1026617, DOI 10.1007/BF01308670 W. De Melo, One-dimensional dynamics, IMPA, Rio de Janeiro.
- M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings, Studia Math. 67 (1980), no. 1, 45–63. MR 579440, DOI 10.4064/sm-67-1-45-63
- Sheldon Newhouse, On some results of Hofbauer on maps of the interval, Dynamical systems and related topics (Nagoya, 1990) Adv. Ser. Dynam. Systems, vol. 9, World Sci. Publ., River Edge, NJ, 1991, pp. 407–421. MR 1164905
- David Ruelle, Thermodynamic formalism, Encyclopedia of Mathematics and its Applications, vol. 5, Addison-Wesley Publishing Co., Reading, Mass., 1978. The mathematical structures of classical equilibrium statistical mechanics; With a foreword by Giovanni Gallavotti and Gian-Carlo Rota. MR 511655
Additional Information
- © Copyright 1995 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 123 (1995), 2901-2907
- MSC: Primary 58F11; Secondary 28D20, 54H20, 58F03
- DOI: https://doi.org/10.1090/S0002-9939-1995-1277099-4
- MathSciNet review: 1277099