Higher bifurcation currents, neutral cycles and the Mandelbrot set

TL;DR

This paper characterizes the support of the bifurcation measure in the moduli space of degree d rational maps, linking it to neutral cycles and generalizing McMullen's results about the Mandelbrot set.

Contribution

It extends McMullen's theorem to higher dimensions, showing the density of certain boundary copies in bifurcation supports for rational maps.

Findings

Support of bifurcation measure coincides with closure of maps with specified neutral cycles.

Homeomorphic copies of boundary products are dense in bifurcation current support.

Provides sharp dimension estimates for bifurcation current supports.

Abstract

We prove that given any $\theta_1,\ldots,\theta_{2d-2}\in \R\setminus\Z$, the support of the bifurcation measure of the moduli space of degree $d$ rational maps coincides with the closure of classes of maps having $2d-2$ neutral cycles of respective multipliers $e^{2i\pi\theta_1},\ldots,e^{2i\pi\theta_{2d-2}}$. To this end, we generalize a famous result of McMullen, proving that homeomorphic copies of $(\partial \Mand)^{k}$ are dense in the support of the $k^{th}$-bifurcation current $T^k_\bif$ in general families of rational maps, where $\Mand$ is the Mandelbrot set. As a consequence, we also get sharp dimension estimates for the supports of the bifurcation currents in any family.