Topological dynamics

Summary

Topological dynamics is the part of the theory of dynamical systems that studies continuous maps and group actions on topological spaces. It uses only the topology, not a measure or a smooth structure. Its questions concern the long-term behaviour of orbits: recurrence, density of orbits, minimal sets, mixing and the growth of complexity. Typical objects are a compact metric space $X$ with a homeomorphism or a continuous map $T \colon X \to X$. It sits beside ergodic theory, which studies the same questions with a measure in place of a topology.