Periods of a max-type equation

TL;DR

This paper thoroughly analyzes the periodic behavior of a specific max-type recurrence relation, characterizing all possible periods and their associated orbits, and discovering a natural number with unique periodic properties.

Contribution

It provides a complete description of the set of periods and periodic orbits for the max-type equation, including the identification of a natural number with special periodicity properties.

Findings

Complete characterization of the set of periods $ ext{Per}(F_4)$

Description of all associated periodic orbits

Existence of a natural number outside $ ext{Per}(F_4)$ with infinite periodic multiples

Abstract

We consider the max-type equation $$x_{n+4}=\max\{x_{n+3},x_{n+2},x_{n+1},0\}-x_n,$$ with arbitrary real initial conditions. We describe completely its set of periods $\mathrm{Per}(F_4)$, as well as its associate periodic orbits. We also prove that there exists a natural number $N\notin\mathrm{Per}(F_4)$ for which $$\left\{N+m:m\geq 1,m\in\mathbb N\right\}\subset\mathrm{Per}(F_4).$$