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Hausdorff dimension of invariant measure of circle diffeomorphisms with a break point

Published online by Cambridge University Press:  07 September 2017

KONSTANTIN KHANIN  and
SAŠA KOCIĆ
KONSTANTIN KHANIN
Affiliation:
Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, ON, Canada M5S 2E4 email [email protected]
SAŠA KOCIĆ
Affiliation:
Department of Mathematics, University of Mississippi, PO Box 1848, University, MS 38677-1848, USA email [email protected]
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Abstract

We prove that, for almost all irrational $\unicode[STIX]{x1D70C}\in (0,1)$, the Hausdorff dimension of the invariant measure of a $C^{2+\unicode[STIX]{x1D6FC}}$-smooth $(\unicode[STIX]{x1D6FC}\in (0,1))$ circle diffeomorphism with a break of size $c\in \mathbb{R}_{+}\backslash \{1\}$, with rotation number $\unicode[STIX]{x1D70C}$, is zero. This result cannot be extended to all irrational rotation numbers.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 5 , May 2019 , pp. 1331 - 1339
Copyright
© Cambridge University Press, 2017 

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