Dimensional characteristics of invariant measures for circle diffeomorphisms
Victoria SadovskayaDepartment of Mathematics and Statistics, University of South Alabama, Mobile, AL 36688, USA
2009en
ABI
Abstract
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.
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