The shadowing property for piecewise monotone interval maps

TL;DR

This paper proves that transitive piecewise monotone interval maps, including β-transformations, do not have the shadowing property, highlighting a fundamental limitation in their dynamical behavior.

Contribution

The paper provides a concise proof that these systems lack shadowing, extending previous results and clarifying their dynamical properties.

Findings

Transitive piecewise monotone interval maps lack shadowing.

Includes β-transformations as a key example.

Extends previous work by Chen and Portela.

Abstract

The property of shadowing has been shown to be fundamental in both the theory of symbolic dynamics as well as continuous dynamical systems. A quintessential class of discontinuous dynamical systems are those driven by transitive piecewise monotone interval maps and in particular $\beta$-transformations, namely transformations of the form $T_{\beta, \alpha} : x \mapsto \beta x + \alpha \; (\operatorname{mod} \, 1)$ acting on $[0,1]$. We provide a short elegant proof showing that this class of dynamical systems does not possess the property of shadowing, complementing and extending the work of Chen and Portela.