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Ergodic properties of bimodal circle maps

Published online by Cambridge University Press:  27 November 2017

SYLVAIN CROVISIER ,
PABLO GUARINO  and
LIVIANA PALMISANO
SYLVAIN CROVISIER
Affiliation:
CNRS - Université Paris-Sud, Laboratoire de Mathématiques d’Orsay, France email [email protected]
PABLO GUARINO
Affiliation:
Instituto de Matemática e Estatística, Universidade Federal Fluminense, Brazil email [email protected]
LIVIANA PALMISANO
Affiliation:
IMPAN, Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland email [email protected]
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Abstract

We give conditions that characterize the existence of an absolutely continuous invariant probability measure for a degree one $C^{2}$ endomorphism of the circle which is bimodal, such that all its periodic orbits are repelling, and such that both boundaries of its rotation interval are irrational numbers. Those conditions are satisfied when the boundary points of the rotation interval belong to a Diophantine class. In particular, they hold for Lebesgue almost every rotation interval. By standard results, the measure obtained is a global physical measure and it is hyperbolic.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 6 , June 2019 , pp. 1462 - 1500
Copyright
© Cambridge University Press, 2017 

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