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The Mañé–Conze–Guivarc’h lemma for intermittent maps of the circle

Published online by Cambridge University Press:  03 February 2009

IAN D. MORRIS
IAN D. MORRIS*
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK (email: Ian.Morris@warwick.ac.uk)
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Abstract

We study the existence of solutions g to the functional inequality f≤g ∘T−g+β, where f is a prescribed continuous function, T is a weakly expanding transformation of the circle having an indifferent fixed point, and β is the maximum ergodic average of f. Using a method due to T. Bousch, we show that continuous solutions g always exist when the Hölder exponent of f is close to 1. In the converse direction, we construct explicit examples of continuous functions f with low Hölder exponent for which no continuous solution g exists. We give sharp estimates on the best possible Hölder regularity of a solution g given the Hölder regularity of f.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 5 , October 2009 , pp. 1603 - 1611
Copyright
Copyright © Cambridge University Press 2009

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