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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… Click to show full 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.
Journal Title: Ergodic Theory and Dynamical Systems
Year Published: 2017
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