Ergodic properties of bimodal circle maps

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 nu...

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Bibliographic Details
Published inErgodic theory and dynamical systems Vol. 39; no. 6; pp. 1462 - 1500
Main Authors CROVISIER, SYLVAIN, GUARINO, PABLO, PALMISANO, LIVIANA
Format Journal Article
LanguageEnglish
Published Cambridge, UK Cambridge University Press 01.06.2019
Cambridge University Press (CUP)
Subjects
Dynamical Systems
Irrationality
Mathematics
Orbits
Original Article
Rotation
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Summary: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.
ISSN:0143-3857
1469-4417
DOI:10.1017/etds.2017.80