Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-27T11:30:47.004Z Has data issue: false hasContentIssue false
  • English
  • Français

Equilibrium states for piecewise monotonic transformations

Published online by Cambridge University Press:  13 August 2009

Franz Hofbauer  and
Gerhard Keller
Franz Hofbauer
Affiliation:
Institut für Mathematik, Universität Wien, Strudlhofgasse 4, A-1090 Wien
Gerhard Keller
Affiliation:
Institut für Angewandte Mathematik, Universität Heidelberg, Im Neuenheimer Feld 294, D-6900 Heidelberg
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We show that equilibrium states μ of a function φ on ([0,1], T), where T is piecewise monotonic, have strong ergodic properties in the following three cases:

(i) sup φ — inf φ <htop(T) and φ is of bounded variation.

(ii) φ satisfies a variation condition and T has a local specification property.

(iii) φ = —log |T′|, which gives an absolutely continuous μ, T is C2, the orbits of the critical points of T are finite, and all periodic orbits of T are uniformly repelling.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 2 , Issue 1 , March 1982 , pp. 23 - 43
Copyright
Copyright © Cambridge University Press 1982

References

REFERENCES

[1]Denker, M., Grillenberger, C. & Sigmund, K.. Ergodic Theory on Compact Spaces. Lecture Notes in Math. no. 527. Springer: Berlin, 1976.CrossRefGoogle Scholar
[2]Dunford, N. & Schwartz, J. T.. Linear Operators I. Interscience: New York, 1958.Google Scholar
[3]Hofbauer, F. & Keller, G.. Ergodic properties of invariant measures for piecewise monotonic transformations. Math. Zeitsch. 180 (1982), 119–140.CrossRefGoogle Scholar
[4]Hofbauer, F.. On intrinsic ergodicity of piecewise monotonic transformations with positive entropy. Israel J. Math. 34 (1979), 213–237.CrossRefGoogle Scholar
[5]Hofbauer, F.. On intrinsic ergodicity of piecewise monotonic transformations with positive entropy II. Israel. J. Math. 38 (1981), 107–115.CrossRefGoogle Scholar
[6]Hofbauer, F.. The structure of piecewise monotonic transformations. Ergod. Th. & Dynam. Sys. 1 (1981), 159–178.CrossRefGoogle Scholar
[7]Hofbauer, F.. Maximal measures for piecewise monotonically increasing transformations on [0,1]. Ergodic Theory. Lecture Notes in Math. no. 729, pp. 66–77. Springer: Berlin, 1979.CrossRefGoogle Scholar
[8]Hofbauer, F.. Examples for the nonuniqueness of the equilibrium state. Trans. Amer. Math. Soc. 228 (1977), 223–241.CrossRefGoogle Scholar
[9]Misiurewicz, M.. Absolutely continuous measures for certain Markov maps of an interval. (Preprint IHES.)Google Scholar
[10]Misiurewicz, M.. Absolutely continuous measures for certain maps of an interval. Publ. Math. IHES. 53 (1981), 17–51.CrossRefGoogle Scholar
[11]Szlenk, W.. Some dynamical properties of certain differentiable mappings of an interval I. Bol. Soc. Math. Mexico. 24 (1979), 57–82.Google Scholar
[12]Szlenk, W.. Some dynamical properties of certain differentiable mappings of an interval II. (Preprint IHES.)Google Scholar
[13]Walters, P.. Equilibrium states for β-transformations and related transformations. Math. Z. 159 (1978), 65–88.CrossRefGoogle Scholar