A generic dimensional property of the invariant measures for circle diffeomorphisms

TL;DR

This paper demonstrates that for any Liouville rotation number, the Hausdorff dimension of invariant measures for circle diffeomorphisms is generically zero, revealing a universal property across smooth dynamics.

Contribution

It establishes a generic nullity of the Hausdorff dimension of invariant measures for circle diffeomorphisms with Liouville rotation numbers, a new insight into their fractal properties.

Findings

Hausdorff dimension of invariant measures is generically zero

Results hold for all Liouville rotation numbers

Applies to the space of smooth, orientation-preserving circle diffeomorphisms

Abstract

Given any Liouville number $\alpha$, it is shown that the nullity of the Hausdorff dimension of the invariant measure is generic in the space of the orientation preserving $C^\infty$ diffeomorphisms of the circle with rotation number $\alpha$.