Global Orbit Patterns for Dynamical Systems On Finite Sets

TL;DR

This paper investigates the structure of all periodic orbits in finite discrete dynamical systems, providing formulas to count global orbit patterns and exploring specific function subsets with unique orbit behaviors.

Contribution

It introduces a precise description of global orbit patterns and derives formulas for counting them, along with computational analysis of special function subsets.

Findings

Formulas for counting global orbit patterns

Identification of interesting orbit patterns in locally rigid functions

Computational enumeration of specific function subsets

Abstract

In this paper, the study of the global orbit pattern (gop) formed by all the periodic orbits of discrete dynamical systems on a finite set $X$ allows us to describe precisely the behaviour of such systems. We can predict by means of closed formulas, the number of gop of the set of all the function from $X$ to itself. We also explore, using the brute force of computers, some subsets of locally rigid functions on $X$, for which interesting patterns of periodic orbits are found.