Fixed Point Shifts of Inert Involutions

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.