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  "body": "\n## Summary\n\nTopological dynamics is the part of the theory of [[dynamical systems|BBZP1R872N]] that studies continuous maps and group actions on topological spaces.\nIt uses only the topology, not a measure or a smooth structure.\nIts questions concern the long-term behaviour of orbits: recurrence, density of orbits, minimal sets, mixing and the growth of complexity.\nTypical objects are a compact metric space $X$ with a [[homeomorphism|0QT00PB5G2]] or a continuous map $T \\colon X \\to X$.\nIt sits beside ergodic theory, which studies the same questions with a measure in place of a topology.\n\n## Explanations\n\n### L1 human\n\nImagine a space of states and a rule that moves each state to the next one, where nearby states move to nearby states.\nTopological dynamics asks qualitative questions about this motion without measuring anything.\nDoes a point come back near where it started?\nDoes some orbit pass near every point?\nFor a rotation of a circle by an irrational angle, every orbit comes back arbitrarily close to its start and passes near every point of the circle.\n",
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