Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-26T00:39:42.436Z Has data issue: false hasContentIssue false
  • English
  • Français

Properties of invariant measures for piecewise expanding one-dimensional transformations with summable oscillations of derivative

Published online by Cambridge University Press:  19 September 2008

Pawel Góra
Pawel Góra
Affiliation:
Department of Mathematics, Concordia University, 7141 Sherbrooke St. W, Montreal, CanadaH4B 1R6
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the properties of absolutely continuous invariant measures for one-dimensional transformations satisfying Schmitt's condition. We prove existence, constrictiveness of the induced Perron-Frobenius operator, openness of supports, estimate on a number of ergodic components, exponential decay of correlation, Bernoulli property, etc.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 14 , Issue 3 , September 1994 , pp. 475 - 492
Copyright
Copyright © Cambridge University Press 1994

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[D-S]Dunford, N. & Schwartz, J. T.. Linear Operators, Part I. Wiley Interscience (Wiley Classics Library): Chichester, 1988.Google Scholar
[G-B]Góra, P. & Boyarsky, A.. Approximating the invariant densities of transformations with infinitely many pieces on the interval. Proc. Amer. Math. Soc. 105 (1989), 922–928.CrossRefGoogle Scholar
[H-K]Hofbauer, F. & Keller, G.. Ergodic properties of invariant measures for piecewise monotonic transformations. Math. Z. 180 (1982), 119–140.Google Scholar
[Kell]Keller, G.. Propriètés ergodiques des endomorphismes dilatants, C 2 par morceaux, des régions bornées du plan. Thèse. Université de Rennes, 1979.Google Scholar
[Kell]Keller, G.. Piecewise monotonic transformations and exactness. Preprint.Google Scholar
[L-L-Y]Lasota, A., Li, T. Y. & Yorke, J. A.. Asymptotic periodicity of the iterates of Markov operators. Trans. Amer. Math. Soc. 286 (1984), 751–764.Google Scholar
[L-Y]Lasota, A. & Yorke, J. A.. On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc. 186 (1973), 481–488.CrossRefGoogle Scholar
[Orn]Ornstein, D.S.. Ergodic Theory, Randomness and Dynamical Systems. Yale University Press: New Haven and London, 1974.Google Scholar
[Rych1]Rychlik, M.R.. Bounded variation and invariant measures. Studia Math. 76 (1983), 69–80.CrossRefGoogle Scholar
[Rych2]Rychlik, M.R.. Invariant measures and the variational principle for Lozi mappings. PhD Thesis. Berkeley, 1983.Google Scholar
[Sch]Schmitt, B.. Contributions a 1'étude de systèmes dynamiques unidimensionnels en théorie ergodique. Thèse. Université de Bourgogne, Dijon, 1986.Google Scholar