On the dynamics of the combinatorial model of the real line

TL;DR

This paper investigates the dynamics of systems on a combinatorial model of the real line, revealing differences in periodic behavior compared to classical models, especially regarding the existence of certain periods.

Contribution

It demonstrates the absence of period-3 points with single-valued maps and shows greater flexibility in period existence using multivalued maps within this combinatorial framework.

Findings

No period-3 points with single-valued maps.

Multivalued maps allow more diverse periods.

Differences from classical Sharkovski's ordering.

Abstract

We study dynamical systems defined on the combinatorial model of the real line. We prove that using single-valued maps there are no periodic points of period 3, which contrasts with the classical and less restrictive setting. Then, we use Vietoris-like multivalued maps to show that there is more flexibility, at least in terms of periods, in this combinatorial framework than in the usual one because we do not have the conditions about the existence of periods given by the Sharkovski Theorem.