## 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 → 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 satisfying the Mittag-Leffler condition and consisting 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 satisfying the Mittag-Leffler condition and consisting 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 ( f, X) is a factor of the inverse limit of a sequence satisfying the Mittag-Leffler condition and consisting of shifts of finite type by a quotient that almost lifts pseudo-orbits.

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 → 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 satisfying the Mittag-Leffler condition and consisting 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 satisfying the Mittag-Leffler condition and consisting 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 ( f, X) is a factor of the inverse limit of a sequence satisfying the Mittag-Leffler condition and consisting of shifts of finite type by a quotient that almost lifts pseudo-orbits.

Original language | English |
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Pages (from-to) | 715–736 |

Number of pages | 22 |

Journal | Inventiones Mathematicae |

Volume | 220 |

DOIs | |

Publication status | Published - 12 Dec 2019 |

## Keywords

- discrete dynamical system
- inverse limits
- pseudo-orbit tracing
- shadowing
- shifts of finite type

## ASJC Scopus subject areas

- General Mathematics