Digraph Representations Of Rational Functions Over $p$-adic Numbers

TL;DR

This paper introduces a graph-based framework for analyzing $p$-adic dynamical systems defined by rational functions, focusing on properties like measure-preservation, invertibility, and ergodicity.

Contribution

It develops a novel digraph representation to characterize key dynamical properties of rational functions over $p$-adic numbers.

Findings

Graph conditions characterize measure-preserving and ergodic behavior

Criteria for invertibility and isometry in $p$-adic rational functions

Conditions for minimality on invariant subsets

Abstract

In this paper, we construct a digraph structure on $p$-adic dynamical systems defined by rational functions. We study the conditions under which the functions are measure-preserving, invertible and isometric, ergodic, and minimal on invariant subsets, by means of graph theoretic properties.