Activity measures of dynamical systems over non-archimedean fields

2019/01/04 by Reimi Irokawa, Irokawa, Reimi
Mathematics · Physics and Astronomy · #Dynamical Systems (math.DS) #FOS: Mathematics #Geometry and complex manifolds #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.1901.01075

openalex publication_date 2019/01/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Toward the understanding of bifurcation phenomena of dynamics on the Berkovich projective line 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.

Citations

• The Dynamical Andre-Oort Conjecture for cubic polynomials
• A non-archimedean λ-lemma

Related

• On a characterization of polynomials among rational functions in non-archimedean dynamics
• Resultant measures and minimal resultant loci for non-archimedean polynomial dynamics
• A dynamical system over a non-archimedean field
• Non-Archimedean techniques and dynamical degenerations
• An a priori bound of rational functions on the Berkovich projective line