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Conjugation between circle maps with several break points

Published online by Cambridge University Press:  21 July 2015

ABDELHAMID ADOUANI
ABDELHAMID ADOUANI*
Affiliation:
University of Carthage, Faculty of Science of Bizerte, Department of Mathematics, Jarzouna 7021, Tunisia email arbi.abdelhamid@gmail.com
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Abstract

Let $f$ and $g$ be two class $P$ -homeomorphisms of the circle $S^{1}$ with break point singularities, which are differentiable maps except at some singular points where the derivative has a jump. Assume that $f$ and $g$ have irrational rotation numbers and the derivatives $\text{Df}$ and $\text{Dg}$ are absolutely continuous on every continuity interval of $\text{Df}$ and $\text{Dg}$ , respectively. We prove that if the product of the $f$ -jumps along all break points of $f$ is distinct from that of $g$ then the homeomorphism $h$ conjugating $f$ and $g$ is a singular function, i.e. it is continuous on $S^{1}$ , but $\text{Dh}(x)=0$  almost everywhere with respect to the Lebesgue measure. This result generalizes previous results for one and two break points obtained by Dzhalilov, Akin and Temir, and Akhadkulov, Dzhalilov and Mayer. As a consequence, we get in particular Dzhalilov–Mayer–Safarov’s theorem: if the product of the $f$ -jumps along all break points of $f$ is distinct from $1$ , then the invariant measure $\unicode[STIX]{x1D707}_{f}$ is singular with respect to the Lebesgue measure.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 36 , Issue 8 , December 2016 , pp. 2351 - 2383
Copyright
© Cambridge University Press, 2015 

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