This thesis is comprised primarily of two separate portions. In the first portion, we exhibit, for any sparse enough increasing sequence of integers {p_n}, a totally minimal, totally uniquely ergodic, and topologically mixing system (X,T) and a continuous function f on X for which the ergodic averages along {p_n} fail to converge for a residual set in X, answering negatively an open question of Bergelson. We also construct a totally minimal, totally uniquely ergodic, and topologically mixing system (X',T') and x' a point in X' so that x' is not a limit point of {T^(p_n)(x')}.
In the second portion, we study perturbations of multidimensional shifts of finite type. Given any Z^d shift of finite type X for d>1 and any word w in the language of X, denote by X_w the set of elements of X in which w does not appear. If X satisfies a uniform mixing condition called strong irreducibility, we obtain exponential upper and lower bounds on the difference in the topological entropies of X and X_w dependent only on the size of w. This result generalizes a result of Lind about Z shifts of finite type.