Iterates of mappings which are almost continuous and open

TL;DR

This paper investigates the behavior of iterates of mappings that are nearly continuous and open, focusing on piecewise monotone functions on intervals or finite graphs, providing a new approach for such analyses.

Contribution

It introduces an approach to study iterates of mappings that are almost continuous and open, especially for piecewise monotone functions on intervals or graphs.

Findings

Applicable to piecewise monotone mappings on intervals or graphs

Provides a new method for analyzing iterates of almost continuous and open mappings

Enhances understanding of the dynamics of such mappings

Abstract

This note presents an approach to studying the iterates of a mapping whose restriction to the complement of a finite set is continuous and open. The main examples to which the approach can be applied are piecewise monotone mappings defined on an interval or a finite graph.