Strong bifurcation loci of full Hausdorff dimension

TL;DR

This paper proves that the support of the bifurcation measure in the moduli space of degree d rational maps has maximal Hausdorff dimension, indicating a rich and complex structure of bifurcations.

Contribution

It establishes that the support of the bifurcation measure has maximal Hausdorff dimension in the moduli space of rational maps, revealing the complexity of bifurcation loci.

Findings

Support of bifurcation measure has maximal Hausdorff dimension 2(2d-2).

Set of maps with 2d-2 neutral cycles is dense in a full Hausdorff dimension set.

Bifurcation loci exhibit maximal fractal complexity.

Abstract

In the moduli space $\mathcal{M}_d$ of degree $d$ rational maps, the bifurcation locus is the support of a closed $(1,1)$ positive current $T_\bif$ which is called the bifurcation current. This current gives rise to a measure $\mu_\bif:=(T_\bif)^{2d-2}$ whose support is the seat of strong bifurcations. Our main result says that $\supp(\mu_\bif)$ has maximal Hausdorff dimension $2(2d-2)$. As a consequence, the set of degree $d$ rational maps having $2d-2$ distinct neutral cycles is dense in a set of full Hausdorff dimension.