Rational Dynamical Systems

TL;DR

This paper introduces rational dynamical systems, extending classical topological dynamics, and proves multiple recurrence results with applications across topology, combinatorics, diophantine approximation, and number theory.

Contribution

It extends classical recurrence results to rational dynamical systems using a new partition theorem for rational numbers.

Findings

Proves multiple recurrence results for rational dynamical systems.

Extends classical recurrence theorems by Furstenberg and Weiss.

Provides applications to various mathematical fields.

Abstract

We introduce the notion of a rational dynamical system extending the classical notion of a topological dynamical system and we prove (multiple) recurrence results for such systems via a partition theorem for the rational numbers proved by Farmaki and the author. In particular, we extend classical recurrence results developed by Furstenberg and Weiss. Also, we give some applications of these topological recurrence results to topology, to combinatorics, to diophantine approximations and to number theory.