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Dimensional characteristics of invariant measures for circle diffeomorphisms

Published online by Cambridge University Press:  03 February 2009

VICTORIA SADOVSKAYA
VICTORIA SADOVSKAYA*
Affiliation:
Department of Mathematics and Statistics, University of South Alabama, Mobile, AL 36688, USA (email: sadovska@jaguar1.usouthal.edu)
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Abstract

We consider pointwise, box, and Hausdorff dimensions of invariant measures for circle diffeomorphisms. We discuss the cases of rational, Diophantine, and Liouville rotation numbers. Our main result is that for any Liouville number τ there exists a C∞ circle diffeomorphism with rotation number τ such that the pointwise and box dimensions of its unique invariant measure do not exist. Moreover, the lower pointwise and lower box dimensions can equal any value 0≤β≤1.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 6 , December 2009 , pp. 1979 - 1992
Copyright
Copyright © Cambridge University Press 2009

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