Topological transitivity

Summary

A continuous map is topologically transitive when it carries every nonempty open set, at some time, into every other nonempty open set. On a compact metric space without isolated points this is the same as having a point whose orbit is dense. A transitive system cannot be split into two open invariant pieces, so it is the topological analogue of an ergodic transformation. Transitivity is one of the three conditions in Devaney's definition of chaos.