Linear Fractional p-Adic and Adelic Dynamical Systems

TL;DR

This paper explores the combined real and p-adic dynamics of linear fractional systems using an adelic framework, revealing properties of rational fixed points and their p-adic behavior.

Contribution

It introduces an adelic approach to analyze linear fractional dynamical systems over real and p-adic fields, highlighting fixed point behaviors and norm relations.

Findings

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

Only finitely many p-adic cases show attractive or repelling fixed points.

Real and p-adic norms of rational fixed points are linked by adelic product formula.

Abstract

Using an adelic approach we simultaneously consider real and p-adic aspects of dynamical systems whose states are mapped by linear fractional transformations isomorphic to some subgroups of GL (2, Q), SL (2, Q) and SL (2, Z) groups. In particular, we investigate behavior of these adelic systems when fixed points are rational. It is shown that any of these rational fixed points is p-adic indifferent for all but a finite set of primes. Thus only for finite number of p-adic cases a rational fixed point may be attractive or repelling. It is also shown that real and p-adic norms of any nonzero rational fixed point are connected by adelic product formula.