Analysis

Learn Ergodic & Homogeneous Dynamics

Long-run behavior of measure-preserving systems: recurrence, ergodic theorems, the Gauss map, and the space of lattices.

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What you'll learn

28 lessons in Ergodic & Homogeneous Dynamics

Measure-preserving systemsPoincaré recurrenceErgodicity & the ergodic theoremsUnique ergodicityMixingThe Koopman operatorThe Gauss mapThe space of latticesGeodesic & horocycle flowsRatner's theoremsThree-gap theorem via latticesEntropy of dynamical systemsFurstenberg recurrence & SzemerédiErgodic decompositionDiscrete spectrum & Halmos–von NeumannKingman's subadditive ergodic theoremOseledets & Lyapunov exponentsTopological entropy & the variational principlePesin theory & SRB measuresRigidity: Furstenberg's ×2, ×3 conjectureWeak mixing & the spectral hierarchySymbolic dynamics & subshiftsBernoulli shifts & Ornstein theoryHyperbolic dynamics: Anosov & Axiom AThermodynamic formalismTopological dynamics & minimalityJoinings & disjointnessSarnak's Möbius disjointness conjecture
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Every idea is taught with motivation and a worked example before the drills, and an FSRS spaced-repetition engine schedules each review for the moment just before you'd forget it. A short placement check finds what you already know, so you start Ergodic & Homogeneous Dynamics exactly where it's useful.

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