On the Hausdorff dimension of invariant measures of piecewise smooth circle homeomorphisms

TL;DR

This paper proves that, under generic conditions, the invariant measure of certain piecewise smooth circle homeomorphisms with irrational rotation number has zero Hausdorff dimension, extending classical rotation number concepts to GIETs.

Contribution

It establishes the zero Hausdorff dimension result for invariant measures of generic piecewise smooth circle homeomorphisms with irrational rotation number, using generalized interval exchange transformations.

Findings

Invariant measure has zero Hausdorff dimension in generic cases.

Extension of rotation number concept to GIETs.

Application to piecewise linear circle homeomorphisms.

Abstract

We show that, generically, the unique invariant measure of a sufficiently regular piecewise smooth circle homeomorphism with irrational rotation number and zero mean nonlinearity (e.g., piecewise linear) has zero Hausdorff dimension. To encode this generic condition, we consider piecewise smooth homeomorphisms as generalized interval exchange transformations (GIETs) of the interval and rely on the notion of combinatorial rotation number for GIETs, which can be seen as an extension of the classical notion of rotation number for circle homeomorphisms to the GIET setting.