This thesis addresses several questions in symbolic dynamics. These involve the image of the dimension representation of a shift of finite type (SFT), the fixed point shifts of involutions of SFTs, and the conjugacy classes of orbit quotients of involutions of SFTs.

We present the first class of examples of mixing SFTs for which the dimension representation is surjective necessarily using nonelementary conjugacies.

Given a mixing shift of finite type X, we consider what subshifts of finite type Y \subset X can be realized as the fixed point shift of an inert involution. 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. If X is the 2-shift, we show that this condition is sufficient to realize Y as the fixed point shift of an involution, up to shift equivalence on X. Given an involution f on X, we characterize what f-invariant subshifts can be realized as the fixed point shift of an involution.

Given a prime p, we classify the conjugacy classes of quotients of 1-sided mixing SFTs which admit free Z/p actions. Finally, given p prime, and X_A a 1-sided mixing SFT, we classify the topological dynamical systems which arise as the orbit quotient systems for a free Z/p action on X_A.