Compactification of a map which is mapped to itself
  We prove that if $T: X \to X$ is a selfmap of a set $X$ such that $\bigcap \{T^{n}X: n\in N}\}$ is a one-point set, then the set $X$ can be endowed with a compact Hausdorff topology so that $T$ is continuous. 
A. Iwanik, L. Janos and F. A. Smith