Some counterexamples in topological dynamics

In this paper, we exhibit, for any sparse-enough increasing sequence {p(n)} of integers, totally minimal, totally uniquely ergodic, and topologically mixing systems (X, T) and (X', V') and f epsilon C(X) for which the averages (1/N Sigma(N-1)(n=0) f (T (pn) x) fail to converge on a residual set in X, and where there exists X' epsilon X' with x' is not an element of {T' p(n) x'}.