Ergodicity and periodic orbits of $p$-adic $(1,2)$-rational dynamical systems with two fixed points

TL;DR

This paper analyzes the dynamics of specific $p$-adic rational functions with two fixed points, classifying their fixed points, invariant sets, and periodic orbits, and examining their ergodic properties.

Contribution

It provides a detailed classification of $p$-adic $(1,2)$-rational functions with two fixed points, including their conjugacy, fixed point types, and periodic orbits, which was not previously established.

Findings

Classification of fixed points and their types

Identification of Siegel disks and basins of attraction

Description of 2- and 3-periodic orbits and ergodicity properties

Abstract

We consider $(1,2)$-rational functions given on the field of $p$-adic numbers $\mathbb Q_p$. In general, such a function has four parameters. We study the case when such a function has two fixed points and show that when there are two fixed points then $(1,2)$-rational function is conjugate to a two-parametric $(1,2)$-rational function. Depending on these two parameters we determine the type of the fixed points, find Siegel disks and the basin of attraction of the fixed points. Moreover, we classify invariant sets and study the ergodicity properties of the function on each invariant set. We describe 2- and 3-periodic orbits of the $p$-adic dynamical systems generated by the two-parametric $(1,2)$-rational functions.