Locating critical points attracted to p-adic attracting cycles

TL;DR

This paper explores the behavior of attracting cycles in non-Archimedean dynamics, establishing conditions under which they attract critical points, contrasting with classical complex dynamics results.

Contribution

It provides a precise criterion based on the multiplier for attracting cycles to attract critical points in non-Archimedean rational maps.

Findings

Identifies a sharp condition on the multiplier for attraction of critical points

Contrasts non-Archimedean dynamics with classical complex dynamics

Enhances understanding of critical point behavior in p-adic dynamics

Abstract

In complex dynamics, a fundamental result of Fatou and Julia asserts that every attracting cycle of a rational map attracts a critical point. The analogous statement fails in non-Archimedean dynamics. For a non-Archimedean rational map, this paper establishes a sharp condition on the multiplier of an attracting cycle ensuring it attracts a critical point.