Then a self-map of  preserves  if and only if it is a bijection, and it is ergodic if and only if  has only one orbit (that is, for every  there exists  such that ).
A self-map preserves itself if and only if it is an isomorphism, and it is ergodic if and only if it has a single trajectory (that is, for every point in the space, there exists another point such that they are connected by a unique path).