Ergodic properties of bimodal circle maps Article - 2019

Sylvain Crovisier, Pablo Guarino, Liviana Palmisano

Sylvain Crovisier, Pablo Guarino, Liviana Palmisano, « Ergodic properties of bimodal circle maps  », Ergodic Theory and Dynamical Systems, 2019, pp. 1462-1500. ISSN 0143-3857

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. The measure obtained is a global physical measure, and it is hyperbolic.

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