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 map preserves itself if and only if it is a reversible transformation, and it is considered ergodic if its paths converge to a single outcome, meaning that for any given initial condition, there exists another state that can be reached through a singular sequence of operations.