# Shifts of finite type as fundamental objects in the theory of shadowing

## TL;DR

This paper establishes a fundamental link between shadowing in dynamical systems and inverse limits of shifts of finite type, providing a characterization that applies to compact totally disconnected spaces and general compact metric spaces.

## Contribution

It proves that shadowing is equivalent to being an inverse limit of shifts of finite type, extending the understanding of shadowing in various topological settings.

## Key findings

- Shadowing systems are conjugate to inverse limits of shifts of finite type.
- In the Cantor set case, shadowing corresponds to inverse limits of shifts of finite type.
- For general compact metric spaces, shadowing systems are factors of inverse limits of shifts of finite type.

## Abstract

Shifts of finite type and the notion of shadowing, or pseudo-orbit tracing, are powerful tools in the study of dynamical systems. In this paper we prove that there is a deep and fundamental relationship between these two concepts. Let $X$ be a compact totally disconnected space and $f:X\to X$ a continuous map. We demonstrate that $f$ has shadowing if and only if the system $({f},{X})$ is (conjugate to) the inverse limit of a directed system of shifts of finite type. In particular, this implies that, in the case that $X$ is the Cantor set, $f$ has shadowing if and only if $(f,X)$ is the inverse limit of a sequence of shifts of finite type. Moreover, in the general compact metric case, where $X$ is not necessarily totally disconnected, we prove that $f$ has shadowing if and only if $({f},{X})$ is a factor of (i.e. semi-conjugate to) the inverse limit of a sequence of shifts of finite type by a quotient that almost lifts pseudo-orbits.