On ergodic behavior of $p$-adic dynamical systems

TL;DR

This paper investigates the ergodic properties of monomial dynamical systems over p-adic numbers, revealing that such systems are ergodic on certain circles around 1, with results depending on the exponent n.

Contribution

It extends the understanding of ergodic behavior from complex unit circles to p-adic contexts, showing ergodicity depends on the parameter n and occurs on circles around 1.

Findings

Monomial maps are ergodic on specific p-adic circles around 1.

Ergodicity depends on the natural number n.

The p-adic case differs from the complex case where ergodicity is on the unit circle.

Abstract

Monomial mappings, $x\mapsto x^n$, are topologically transitive and ergodic with respect to Haar measure on the unit circle in the complex plane. In this paper we obtain an anologous result for monomial dynamical systems over $p-$adic numbers. The process is, however, not straightforward. The result will depend on the natural number $n$. Moreover, in the $p-$adic case we never have ergodicity on the unit circle, but on the circles around the point 1.