Strong bifurcation loci of full Hausdorff dimension

2011/03/14 by Gauthier, Thomas · 1 citation
#28A78 #32U15 #37F45 #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1103.2656

Abstract

In the moduli space 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 μ_\bif:=(T_\bif)^2d-2 whose support is the seat of strong bifurcations. Our main result says that \supp(μ_\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.

Cited by

• A dynamical approach to studying the Lee-Yang zeros for the Potts Model on the Cayley Tree

Related

• Higher bifurcation currents, neutral cycles and the Mandelbrot set
• Collet, Eckmann and the bifurcation measure
• Perturbations d'exemples de Latt `es et dimension de Hausdorff du lieu\n de bifurcation
• Lyapunov exponents, bifurcation currents and laminations in bifurcation loci
• Connectivity of the branch locus of moduli space of rational maps