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.
Shadowing property and transitivity of a set-valued map and its inverse limit
School of Computer Science and Technology, Dongguan University of Technology（东莞理工学院计算机科学与技术学院）；School of Data Science, Zhuhai College of Science and Technology（珠海科技学院数据科学学院）
Characterizations of ω-Limit Sets of Topologically Hyperbolic Systems
University of Bristol（布里斯托大学）；University of Birmingham（伯明翰大学）；AGH University of Science and Technology（AGH科学技术大学）；Institute of Mathematics, Polish Academy of Sciences（波兰科学院数学研究所）；Baylor University（贝勒大学）
