American Institute of Mathematical Sciences

Given a mixing shift of finite type $X$, we consider which subshifts of finite type $Y \subset X$ 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.