The Shadowing Property for Piecewise Monotone Interval Maps

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.