A note on odometers and shadowing

TL;DR

This paper investigates conditions under which points in a compact metric space under a continuous map have orbits that converge to periodic or odometer-like behavior, emphasizing the role of the shadowing property.

Contribution

It establishes a sufficient condition for orbit convergence to periodic or odometer behavior and shows density of such points under the shadowing property.

Findings

Points with orbits converging to periodic or odometer are dense under shadowing.

Provides a sufficient condition for orbit convergence to periodic or odometer.

Shows the importance of shadowing in orbit behavior analysis.

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.