A note on odometers and shadowing
Noriaki Kawaguchi
TL;DR
The paper addresses when orbits under a continuous map $f$ on a compact metric space converge to a periodic orbit or an odometer, formalized via $A(f)=\{x\in X:\;\omega(x,f)\text{ is a periodic orbit or an odometer}\}$. It proves a concrete sufficient condition for membership in $A(f)$ using proximity to an $A(f)$-point, and it shows that if $f$ has the shadowing property then $X$ is dense in $A(f)$, i.e., $X=\overline{A(f)}$. The work leverages a characterization of odometers as infinite minimal sets with all points regularly recurrent and employs omega-limit analysis and shadowing constructions (involving $\delta_j$-pseudo orbits) to connect regular recurrence with odometer dynamics. It additionally relates these results to chain recurrence and chain continuity, highlighting how shadowing enforces regular, simplex-like asymptotic behavior in a broad class of dynamical systems.
Abstract
For a continuous self-map of a compact metric space, we provide a sufficient condition for the orbit of a point to converge to a periodic orbit or an odometer. We show that if a continuous self-map of a compact metric space has the shadowing property, then the set of points whose orbits converge to periodic orbits or odometers is dense.