Global Orbit Patterns for One Dimensional Dynamical Systems

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.