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Non-rigidity for circle homeomorphisms with several break points

Published online by Cambridge University Press:  28 November 2017

ABDELHAMID ADOUANI  and
HABIB MARZOUGUI
ABDELHAMID ADOUANI
Affiliation:
University of Carthage, Faculty of Science of Bizerte, Department of Mathematics, Jarzouna, 7021, Tunisia email [email protected], [email protected], [email protected]
HABIB MARZOUGUI
Affiliation:
University of Carthage, Faculty of Science of Bizerte, Department of Mathematics, Jarzouna, 7021, Tunisia email [email protected], [email protected], [email protected]
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Abstract

In this work, we consider two class $P$-homeomorphisms, $f$ and $g$, of the circle with break point singularities, that are differentiable maps except at some singular points where the derivative has a jump. Assume that they have the same irrational rotation number of bounded type and that the derivatives $\text{Df}$ and $\text{Dg}$ are absolutely continuous on every continuity interval of $\text{Df}$ and $\text{Dg}$, respectively. We show that if $f$ and $g$ are not break-equivalent, then any topological conjugating $h$ between $f$ and $g$ is a singular function, i.e., it is continuous on the circle, but $\text{Dh}(x)=0$ almost everywhere (a.e.) with respect to the Lebesgue measure. In particular, this result holds under some combinatorial assumptions on the jumps at break points. It also generalizes previous results obtained for one and two break points and complements that of Cunha–Smania which was established for break equivalence.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 9 , September 2019 , pp. 2305 - 2331
Copyright
© Cambridge University Press, 2017 

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