Collet, Eckmann and the bifurcation measure

2017/05/17 by Astorg, Matthieu, Gauthier, Thomas, Mihalache, Nicolae +1
#Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1705.06114

Abstract

The moduli space M_d of degree d≥2 rational maps can naturally be endowed with a measure μ_bif detecting maximal bifurcations, called the bifurcation measure. We prove that the support of the bifurcation measure μ_bif has positive Lebesgue measure. To do so, we establish a general sufficient condition for the conjugacy class of a rational map to belong to the support of μ_bif and we exhibit a large set of Collet-Eckmann rational maps which satisfy this condition. As a consequence, we get a set of Collet-Eckmann rational maps of positive Lebesgue measure which are approximated by hyperbolic rational maps.

Related

• Higher bifurcation currents, neutral cycles and the Mandelbrot set
• Strong bifurcation loci of full Hausdorff dimension
• Compactifications and measures for rational maps
• The bifurcation measure is exponentially mixing
• The maximal entropy measure detects non-uniform hyperbolicity