Global Orbit Patterns for One Dimensional Dynamical Systems

TL;DR

This paper analyzes the global orbit patterns of one-dimensional discrete dynamical systems on finite sets, providing formulas for counting these patterns and exploring special subsets with interesting behaviors.

Contribution

It formalizes the concept of global orbit patterns and derives closed-form formulas for their enumeration across all functions on finite sets.

Findings

Derived formulas for counting global orbit patterns.

Identified interesting patterns in special subsets of functions.

Validated formulas through computational experiments.

Abstract

In this article, we study the behaviour of discrete one-dimensional dynamical systems associated to functions on finite sets. We formalise the global orbit pattern formed by all the periodic orbits (gop) as the ordered set of periods when the initial value thumbs the finite set in the increasing order. We are able to predict, using closed formulas, the number of gop for the set $\mathcal{F}_N$ of all the functions on $X$. We also explore by computer experiments special subsets of $\mathcal{F}_N$ in which interesting patterns of gop are found.