AIMS ’ Journals Volume X, Number 0X, XX 200X pp. X–XX FIXED POINT SHIFTS OF INERT INVOLUTIONS

ABSTRACT. Given a mixing shift of finite type X, we consider which subshifts of finite type Y ⊂ X can be realized as the fixed point shift of an inert involution of X. We establish a condition on the periodic points of X and Y that is necessary for Y to be the fixed point shift of an inert involution of X. We show that this condition is sufficient to realize Y as the fixed point shift of an involution, up to shift equivalence on X, if X is a shift of finite type with Artin-Mazur zeta function equivalent to 1 mod 2. Given an inert involution f of a mixing shift of finite type X, we characterize what f-invariant subshifts can be realized as the fixed point shift of an inert involution. 1. Introduction and Statement of Results. We refer the reader to Section 2 for precise definitions. The group of shift commuting homeomorphisms of a shift of finite type X is called the automorphism group (of X) and is denoted as Aut(σX). The group Aut(σX) has been a vital tool in attempts to classify shifts of finite type, but in general, Aut(σX) is large and mysterious. For example, the automorphism group of the 2-shift is countably infinite