Piecewise continuous and monotonic maps on the interval

TL;DR

This paper investigates piecewise continuous, monotonic interval maps with finitely many discontinuities, introducing a generalized concept of periodic orbits and analyzing their properties compared to continuous maps.

Contribution

It defines and studies the notion of closed structure as a generalization of periodic orbits, highlighting differences from continuous maps.

Findings

Introduces the concept of closed structure as a generalization of periodic orbit.

Analyzes properties of periodic orbits away from discontinuities.

Highlights main differences from the continuous case.

Abstract

Let $f$ be a piecewise continuous and monotonic map on the interval with at most finitely many discontinuities and turning points. In this paper we study properties about this class of maps and show its main difference from the continuous case. We define and study the notion of closed structure, which can be seen as an generalization of periodic orbit. We also study the periodic orbits that are away from the discontinuities of $f$, extending the notion of trapped and free orbits.