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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 provethat 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 continuousmap. 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 shiftsof finite type. In particular, this implies that, in the case that $X$ is theCantor set, $f$ has shadowing if and only if $(f,X)$ is the inverse limit of asequence of shifts of finite type. Moreover, in the general compact metriccase, where $X$ is not necessarily totally disconnected, we prove that $f$ hasshadowing 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 thatalmost lifts pseudo-orbits.

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