# $p$-adic dynamical systems of $(2,2)$-rational functions with unique fixed point

## TL;DR

This paper investigates the dynamics of $(2,2)$-rational functions with a unique fixed point over complex $p$-adic fields, revealing properties like indifferent fixed points, Siegel disks, and non-ergodicity on invariant spheres.

## Contribution

It provides explicit descriptions of the fixed points, Siegel disks, limit sets, and ergodic properties of these $p$-adic dynamical systems, which were not previously detailed.

## Key findings

- Fixed point is indifferent, affecting convergence behavior.
- Identified Siegel disks and bounded limit points for trajectories.
- Proved non-ergodicity of the system on invariant spheres.

## Abstract

We consider a family of $(2,2)$-rational functions given on the set of complex $p$-adic field $\mathbb{C}_p$. Each such function has a unique fixed point. We study $p$-adic dynamical systems generated by the $(2,2)$-rational functions. We show that the fixed point is indifferent and therefore the convergence of the trajectories is not the typical case for the dynamical systems. Siegel disks of these dynamical systems are found. We obtain an upper bound for the set of limit points of each trajectory, i.e., we determine a sufficiently small set containing the set of limit points. For each $(2,2)$-rational function on $\mathbb{C}_p$ there are two points $\hat x_1=\hat x_1(f)$, $\hat x_2=\hat x_2(f)\in \mathbb{C}_p$ which are zeros of its denominator. We give explicit formulas of radiuses of spheres (with the center at the fixed point) containing some points such that the trajectories (under actions of $f$) of the points after a finite step come to $\hat x_1$ or $\hat x_2$. Moreover for a class of $(2,2)$-rational functions we study ergodicity properties of the dynamical systems on the set of $p$-adic numbers $Q_p$. For each such function we describe all possible invariant spheres. We show that the $p$-adic dynamical system reduced on each invariant sphere is not ergodic with respect to Haar measure.