Activity measures of dynamical systems over non-archimedean fields

Abstract

Toward the understanding of bifurcation phenomena of dynamics on the Berkovich projective line ${\mathbb{P}}^{1,an}$ over non-archimedean fields, we study the stability (or passivity) of critical points of families of polynomials parametrized by analytic curves. We construct the activity measure of a critical point of a family of rational functions, and study its properties. For a family of polynomials, we study more about the activity locus such as its relation to boundedness locus, i.e., the Mandelbrot set, and to the normality of the sequence of the forward orbit.

16 pages

References in corpus (3)

• Formes différentielles réelles et courants sur les espaces de Berkovich
• Attracting cycles in p-adic dynamics and height bounds for post-critically finite maps
• The Dynamical Andre-Oort Conjecture for cubic polynomials