Sharkovskii order for non-wandering points

TL;DR

This paper extends Sharkovskii's order to non-wandering points using ultrapower techniques, broadening the understanding of periodicity and recurrence in interval maps.

Contribution

It introduces an extension of Sharkovskii's order to non-wandering points via ultrapower methods, generalizing classical periodic point results.

Findings

Extended Sharkovskii order applies to non-wandering points.

Non-wandering points exhibit similar periodicity structures as periodic points.

The approach broadens the scope of Sharkovskii's theorem in dynamical systems.

Abstract

For a map $f:I \rightarrow I$, a point $x \in I$ is periodic with period $p \in \mathbb{N}$ if $f^p(x)=x$ and $f^j(x)\not=x$ for all $0<j<p$. When $f$ is continuous and $I$ is an interval, a theorem due to Sharkovskii (\cite{BC}) states that there is an order in $\mathbb{N}$, say $\lhd$, such that, if $f$ has a periodic point of period $p$ and $p \lhd q$, then $f$ also has a periodic point of period $q$. In this work, we will see how an extension of this order $\lhd$ to an ultrapower of the integer numbers yields a Sharkovskii-type result for non-wandering points of $f$.