Dimensional characteristics of invariant measures for circle diffeomorphisms

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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 there exists a C circle diffeomorphism with rotation number 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.

Original languageEnglish (US)
Pages (from-to)1979-1992
Number of pages14
JournalErgodic Theory and Dynamical Systems
Volume29
Issue number6
DOIs
StatePublished - Dec 1 2009

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

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