Universal functions for the Sharkovsky classes of maps

TL;DR

This paper introduces a universal interval map that encompasses all orbit patterns in the first Sharkovsky class, providing a comprehensive understanding of maps with only fixed points.

Contribution

It constructs and characterizes a universal map for the first Sharkovsky class, extending the concept of universality to this specific class of interval maps.

Findings

Existence of a universal map for the first Sharkovsky class

Characterization of all such universal maps within this class

Application to contractions on interval

Abstract

We exhibit a single interval map (called universal map) that admits all those orbit patterns which are available in the first Sharkovsky class. An interval map is said to be in the first Sharkovsky class if every periodic point of it is a fixed point. This provides a way to find universal maps in the class of contractions on interval. We also characterize all such universal maps in the first Sharkovsky class.