The Shadowing Property for Piecewise Monotone Interval Maps

A. Bura; C. Good; T. Samuel

The Shadowing Property for Piecewise Monotone Interval Maps

A. Bura
, C. Good
, T. Samuel^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

Abstract

The property of shadowing is 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 β-transformations, namely transformations of the form T_β,α : x 7→ βx + α (mod 1) acting on [0, 1]. We provide a short elegant proof showing that this class of dynamical systems do not possess the property of shadowing, complementing and extending the work of Liang Chen and Raquel Portela.

Original language	English
Journal	The American Mathematical Monthly
Publication status	Accepted/In press - 26 Oct 2025

Bibliographical note

Not yet published as of 02/03/2026.

Embargoed Document

BuraA2025Shadowing_AAM
Accepted author manuscript, 134 KB
Licence: Creative Commons: Attribution (CC BY)
Embargo ends: 30/03/26

The shadowing property for piecewise monotone interval maps
Bura, A., Good, C. & Samuel, T., 7 Feb 2025, (E-pub ahead of print) arXiv, 6 p.
Research output: Working paper/Preprint › Preprint
6 Downloads (Pure)

Cite this

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title = "The Shadowing Property for Piecewise Monotone Interval Maps",

abstract = "The property of shadowing is 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 β-transformations, namely transformations of the form Tβ,α : x 7→ βx + α (mod 1) acting on [0, 1]. We provide a short elegant proof showing that this class of dynamical systems do not possess the property of shadowing, complementing and extending the work of Liang Chen and Raquel Portela.",

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AU - Samuel, T.

N1 - Not yet published as of 02/03/2026.

PY - 2025/10/26

Y1 - 2025/10/26

N2 - The property of shadowing is 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 β-transformations, namely transformations of the form Tβ,α : x 7→ βx + α (mod 1) acting on [0, 1]. We provide a short elegant proof showing that this class of dynamical systems do not possess the property of shadowing, complementing and extending the work of Liang Chen and Raquel Portela.

AB - The property of shadowing is 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 β-transformations, namely transformations of the form Tβ,α : x 7→ βx + α (mod 1) acting on [0, 1]. We provide a short elegant proof showing that this class of dynamical systems do not possess the property of shadowing, complementing and extending the work of Liang Chen and Raquel Portela.

UR - https://www.tandfonline.com/journals/uamm20

M3 - Article

SN - 0002-9890

JO - The American Mathematical Monthly

JF - The American Mathematical Monthly

ER -

The Shadowing Property for Piecewise Monotone Interval Maps

Abstract

Bibliographical note

Embargoed Document

Fingerprint

Research output

The shadowing property for piecewise monotone interval maps

Cite this