Dynamical Systems

Dynamics studies systems evolving through iteration or time, including recurrence, stability, and long-term behavior.

Ergodic theory holds just over half of the subject's articles, most of them on the almost-everywhere convergence of ergodic averages: the Birkhoff ergodic theorem for maps preserving a finite measure, whose limit is the conditional expectation on the invariant σ-algebra, Kingman's subadditive ergodic theorem, and Bourgain's theorem along polynomial times. Topological and symbolic dynamics has the Curtis–Hedlund–Lyndon theorem, smooth dynamics the shadowing lemma and the planar entropy formulas of Pesin and Young, and low-dimensional dynamics the Poincaré–Bendixson theorem. In complex dynamics the Mandelbrot set is connected, and by the Fatou–Julia dichotomy the filled Julia set of \(z^2 + c\) is connected for \(c\) in it and a Cantor set otherwise.

• Ergodic theory
• Topological and symbolic dynamics
• Low-dimensional dynamics
• Smooth dynamics
• Complex dynamics

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