2017/02/16 by Good, Chris, Meddaugh, Jonathan
#37B20 #54H20 #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.1702.05170
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 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.