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Singular measures for class P-circle homeomorphisms with several break points

Published online by Cambridge University Press:  30 November 2012

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

Let f be a class P-homeomorphism of the circle with break point singularities, that is, differentiable except at some singular points where the derivative has a jump. Let f have irrational rotation number and Df be absolutely continuous on every continuity interval of Df. We prove that if the product of the f-jumps along any subset of break points is distinct from 1 then the invariant measure μf is singular with respect to the Haar measure. This result generalizes previous results obtained by Dzhalilov and Khanin, Dzhalilov, Akhadkulov, Dzhalilov–Liousse and Mayer. Moreover, we prove that if the rotation number ρ(f) is irrational of bounded type then (a) if the product of the f-jumps on some orbit is distinct from 1 then the invariant measure μf is singular with respect to the Haar measure m, and (b) if the product of the f-jumps on each orbit is equal to 1 and D2f∈Lp (S1) for some p>1 then μfis equivalent to the Haar measure.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 2 , April 2014 , pp. 423 - 456
Copyright
©2012 Cambridge University Press 

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