Hausdorff Dimension of Closure of Cycles in d-Maps on the Circle

TL;DR

This paper investigates the fractal dimensions of cycle closures in circle maps, confirming a conjecture that relates their Hausdorff dimension to the base map and cycle degree.

Contribution

It proves McMullen's conjecture by linking the Hausdorff dimension of cycle closures to their degree and the base map, using combinatorial properties of base-d expansions.

Findings

Hausdorff dimension of cycle closures equals log m / log d

Characterization of invariant finite subsets called cycles

Confirmation of McMullen's conjecture

Abstract

We study the dynamics of the map $x$ to $dx$ (mod 1) on the unit circle. We characterize the invariant finite subsets of this map which are called cycles and are graded by their degrees. By looking at the combinatorial properties of the base-d expansion of the elements in the cycles, we prove a conjecture of Curt McMullen that the Hausdorff dimension of the closure of degree-m cycles is equal to log $m$ / log $d$.