Strong bifurcation loci of full Hausdorff dimension. (Lieux de bifurcation maximale de dimension de Hausdorff totale.) (English. French summary) Zbl 1326.37036

Gauthier, Thomas

Ann. Sci. Éc. Norm. Supér. (4) 45, No. 6, 947-984 (2012).

In the moduli space of degree-\(d\) rational maps, the bifurcation loci is the support of a positive \((1,1)\) positive current \(T\) called the bifurcation current. This current gives rise to a mesaure \(\mu_{\mathrm{bif}}:=(T)^{2d-2}\). The main result of this work states that the support of \(\mu_{\mathrm{bif}}\) 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.

Reviewer: Federico Quallbrunn (Buenos Aires)

MSC:

37F45	Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations (MSC2010)
32U15	General pluripotential theory
28A78	Hausdorff and packing measures

Keywords:

bifurcation loci; Hausdorff dimension; Misiurewicz map