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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
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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

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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