The shadowing property for piecewise monotone interval maps
Adarsh Bura, Chris Good, Tony Samuel
TL;DR
This work addresses whether the shadowing property holds for discontinuous dynamical systems arising from transitive piecewise monotone interval maps and β-transformations. It provides a short, elegant proof that transitive piecewise continuous monotone interval maps do not have finite shadowing, hence no shadowing, by constructing a strategic finite δ-pseudo-orbit near a partition boundary and deriving a contradiction. For β-transformations, it reduces non-transitivity to a transitive first return map that is affine-conjugate to a transitive β-transformation, transferring the non-shadowing conclusion via this conjugacy. The results clarify the limitations of shadowing in these classical intervals maps, contrasting with continuous or subshift-of-finite-type settings, and open avenues for extending the approach to broader piecewise systems and negative β-transformations, with implications for symbolic dynamics and number-theoretic expansions.
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 $β$-transformations, namely transformations of the form $T_{β, α} : x \mapsto βx + α\; (\operatorname{mod} \, 1)$ 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.