Dimensional characteristics of invariant measures for circle diffeomorphisms

Details Dimensional characteristics of invariant measures for circle diffeomorphisms Dimensional characteristics of invariant measures for circle diffeomorphisms

2008-09-02

arxiv.org/abs/0809.0343v1

Mathematics

Dynamical Systems

16 pages

Scientific paper

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 $\tau$ there exists a $C^\infty$ circle diffeomorphism with rotation number $\tau$ 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 $0\le \beta \le 1$.

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