Periodic points and automorphisms of the shift

The automorphism group of a topological Markov shift is studied by way of periodic points and unstable sets. A new invariant for automorphisms of dynamical systems, the gyration function, is used to characterize those automorphisms of finite subsystems of the full shift on n symbols which can be extended to a composition of involutions of the shift. It is found that for any automorphism U of a subshift of finite type S, for all large integers M the map USM is a topological Markov shift whose unstable sets equal those of S. This fact yields, by way of canonical measures and dimension groups, information about dynamical properties of USk such as the zeta function and entropy. © 1987 American Mathematical Society.