On minimal decomposition of $p$-adic homographic dynamical systems

TL;DR

This paper investigates the structure of $p$-adic homographic dynamical systems, especially those without fixed points, showing they can be decomposed into minimal subsystems with known invariant measures.

Contribution

It provides a minimal decomposition for $p$-adic homographic systems lacking fixed points, extending previous results to this case.

Findings

Decomposition into finitely many minimal subsystems

Explicit description of minimal subsystems

Determination of unique invariant measures

Abstract

A homographic map in the field of $p$-adic numbers $\mathbb{Q}_p}$ is studied as a dynamical system on $\mathbb{P}^{1}(\mathbb{Q}_p)$, the projective line over $\mathbb{Q}_p$. If such a system admits one or two fixed points in $\mathbb{Q}_p$, then it is conjugate to an affine dynamics whose dynamical structure has been investigated by Fan and Fares. In this paper, we shall mainly solve the remaining case that the system admits no fixed point. We shall prove that this system can be decomposed into a finite number of minimal subsystems which are topologically conjugate to each other. All the minimal subsystems are exhibited and the unique invariant measure for each minimal subsystem is determined.