p-Adic and Adelic Rational Dynamical Systems

TL;DR

This paper explores the properties of rational dynamical systems using adelic methods, focusing on fixed points and their behavior across real and p-adic contexts, revealing that rational fixed points are generally p-adic indifferent except for finitely many primes.

Contribution

It extends previous work by analyzing the p-adic and adelic properties of linear fractional dynamical systems with rational fixed points, highlighting their typical indifference across primes.

Findings

Rational fixed points are p-adic indifferent for all but finitely many primes.

Finite primes can exhibit attractive or repelling behavior at rational fixed points.

The study provides insights into the p-adic dynamics of rational maps and suggests possible generalizations.

Abstract

In the framework of adelic approach we consider real and p-adic properties of dynamical system given by linear fractional map f (x) = (a x + b)/(c x + d), where a, b, c and d are rational numbers. In particular, we investigate behavior of this adelic dynamical system when fixed points are rational. It is shown that any of rational fixed points is p-adic indifferent for all but a finite set of primes. Only for finite number of p-adic cases a rational fixed point may be attractive or repelling. The present analysis is a continuation of the paper math-ph/0612058. Some possible generalizations are discussed.