Ergodicity properties of $p$ -adic $(2,1)$-rational dynamical systems with unique fixed point

TL;DR

This paper investigates the ergodic behavior of $p$-adic $(2,1)$-rational dynamical systems with a unique fixed point, identifying conditions for ergodicity on invariant spheres within the $p$-adic field.

Contribution

It characterizes invariant spheres and determines ergodicity conditions for $p$-adic $(2,1)$-rational systems with a unique fixed point, a novel analysis in $p$-adic dynamics.

Findings

Identified all invariant spheres for the systems.

Established criteria for ergodicity on these spheres.

Found specific spheres where the system is ergodic.

Abstract

We consider a family of $(2,1)$-rational functions given on the set of $p$-adic field $Q_p$. Each such function has a unique fixed point. We study ergodicity properties of the dynamical systems generated by $(2,1)$-rational functions. For each such function we describe all possible invariant spheres. We characterize ergodicity of each $p$-adic dynamical system with respect to Haar measure reduced on each invariant sphere. In particular, we found an invariant spheres on which the dynamical system is ergodic and on all other invariant spheres the dynamical systems are not ergodic.