$p$-Adic $(3,2)$-rational dynamical systems with three fixed points

TL;DR

This paper analyzes $p$-adic rational dynamical systems with three fixed points, exploring their Siegel disks, basins of attraction, and trajectory behaviors for different parameter values.

Contribution

It provides a detailed study of the fixed points, Siegel disks, and basins in $p$-adic $(3,2)$-rational systems, revealing new dynamical properties.

Findings

Siegel disks may coincide or be disjoint

Basins of attraction are characterized

Existence of trajectories escaping arbitrarily far

Abstract

In this paper we consider dynamical systems generated by $(3,2)$-rational functions on the field of $p$-adic complex numbers. Each such function has three fixed points. We show that Siegel disks of the dynamical system may either coincide or be disjoint for different fixed points. Also, we find the basin of each attractor of the dynamical system. We show that, for some values of the parameters, there are trajectories which go arbitrary far from the fixed points.