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Strong bifurcation loci of full Hausdorff dimension

Authors
  • Gauthier, Thomas
Type
Preprint
Publication Date
Mar 22, 2012
Submission Date
Mar 14, 2011
Source
arXiv
License
Yellow
External links

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.

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