Ergodic properties of bimodal circle maps

TL;DR

This paper establishes conditions under which certain bimodal circle maps possess a unique, hyperbolic, absolutely continuous invariant measure, especially when boundary rotation numbers are irrational and satisfy Diophantine conditions.

Contribution

It provides new criteria for the existence of physical measures in bimodal circle maps with irrational boundary rotation numbers, extending understanding of their ergodic properties.

Findings

Existence of absolutely continuous invariant measures under Diophantine conditions

Conditions satisfied for Lebesgue almost every rotation interval

The invariant measure is hyperbolic and a global physical measure

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.