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↦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.
Fields of Study & Subjects
Mathematical Dynamics and Fractals
Stability and Controllability of Differential Equations
Quantum chaos and dynamical systems
Publication details
- Journal / source
- American Mathematical Monthly
- Publisher
- Informa UK Limited
- Volume
- 133
- Issue
- 8
- Pages
- 1–8
- Publication date
- 16 July 2026
- Language
- EN