# Activity measures of dynamical systems over non-archimedean fields

## TL;DR

This paper investigates the stability and activity of critical points in non-archimedean dynamical systems, constructing measures to understand bifurcations and their relation to classical sets like the Mandelbrot set.

## Contribution

It introduces the activity measure for critical points in non-archimedean dynamics and explores its properties and relation to stability and classical bifurcation loci.

## Key findings

- Constructed the activity measure for critical points in non-archimedean families.
- Analyzed the relation between activity locus and the Mandelbrot set.
- Connected activity measures to the normality of forward orbits.

## 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.