Ergodic theory

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Requisitos previos obligatorios, hasta el fondo: 42

Aprenderás a

• Prove Poincaré recurrence and state the Birkhoff ergodic theorem, interpreting 'time average equals space average'.
• Decide ergodicity and mixing for rotations of the circle, doubling maps and Bernoulli shifts.
• Use the Gauss map to study the statistics of continued fraction digits.

Definición

• A measure-preserving system (X, μ, T) has μ(T⁻¹A) = μ(A) for every measurable A.
• T is ergodic if every measurable A with T⁻¹A = A has μ(A) = 0 or 1.
• T is mixing if μ(T⁻ⁿA ∩ B) → μ(A) μ(B) as n → ∞ for all measurable A, B.

Teorema

• Poincaré recurrence: in a finite measure-preserving system almost every point of a set of positive measure returns to it infinitely often.
• Birkhoff: for ergodic T and f ∈ L¹(μ), (1/n) Σ_{k<n} f(Tᵏx) → ∫ f dμ for almost every x.
• Rotation of the circle by α is ergodic exactly when α is irrational, and never mixing.
• The Gauss measure dx / ((1 + x) log 2) on [0, 1) is invariant for the Gauss map x ↦ {1/x}.

Errores frecuentes

• Thinking ergodic means every orbit is dense or equidistributed; the ergodic theorem holds for almost every point, and exceptional points can exist.
• Believing ergodicity implies mixing; irrational rotations are ergodic but not mixing.

Descubrimientos opcionales

• Extremal and additive combinatorics Furstenberg proved Szemerédi's theorem on arithmetic progressions with an ergodic recurrence theorem, starting a lasting bridge between ergodic theory and combinatorics.
• Analytic number theory Weyl's equidistribution theorem — n α mod 1 is equidistributed for irrational α — is both an ergodic theorem for rotations and a result in number theory.
• ← Dynamical systems and chaos
• ← Extremal and additive combinatorics

Itinerarios de aprendizaje

• Analysis, PDE and dynamics

Lecturas complementarias

• Peter Walters, An Introduction to Ergodic Theory (1982)
• Manfred Einsiedler and Thomas Ward, Ergodic Theory with a View towards Number Theory (2011)