Dynamics of measure-preserving transformations: time averages equal space averages (Birkhoff), structure of mixing, entropy as an invariant, operator-theoretic (Koopman/Perron-Frobenius) viewpoint. Substrate of statistical mechanics,…
ergodic-theory
Measure-preserving transformation
Map T: X → X on a probability space (X, Σ, μ) such that μ(T⁻¹(A)) = μ(A) for every A ∈ Σ. The base object of ergodic theory; all invariants…
Ergodic system
Measure-preserving (X, Σ, μ, T) where every T-invariant set A (i.e., T⁻¹(A) = A) has μ(A) ∈ {0, 1}. Equivalent to: the only T-invariant L²…
Birkhoff pointwise ergodic theorem
For T measure-preserving and f ∈ L¹(μ), the time average (1/n) Σ₀ⁿ⁻¹ f∘T^k converges a.e. to a T-invariant function ƒ*. If T is ergodic, ƒ*…
Von Neumann mean ergodic theorem
L² version of Birkhoff: for isometry U on Hilbert space H, the averages (1/n)Σ_{k<n} U^k converge strongly to the orthogonal projection…
Poincaré recurrence theorem
For a measure-preserving T on a probability space and any A with μ(A) > 0, a.e. point of A returns to A under iteration of T infinitely…
Mixing dynamical system
Strong mixing: lim_{n→∞} μ(A ∩ T⁻ⁿ B) = μ(A)μ(B). Strictly stronger than ergodicity; captures asymptotic statistical independence. Weak…
Kolmogorov–Sinai entropy
Measure-theoretic entropy h(T) of a transformation: supremum over finite partitions P of h(T,P) = lim (1/n) H(P ∨ T⁻¹P ∨ … ∨ T⁻ⁿ⁺¹P).…
Koopman operator
Given T on (X,μ), the induced unitary (or isometry) U_T on L² defined by U_T f = f ∘ T. Lifts measure-preserving dynamics to linear…
Unique ergodicity
A topological system admits exactly one T-invariant probability measure. Implies that every continuous observable's time average converges…
Ratner measure-rigidity theorems
For unipotent flows on homogeneous spaces G/Γ, every invariant ergodic probability measure is algebraic (supported on a closed orbit of a…
Krylov-Bogolyubov theorem
Every continuous self-map T of a compact metrisable space admits a T-invariant Borel probability measure. Proved by Cesàro-averaging any…
Bernoulli shift
Product measure p^ℤ on {0,1}^ℤ together with the left shift σ. Mixing, Kolmogorov, and of Kolmogorov-Sinai entropy H(p). By Ornstein's…
Arnold's cat map
Hyperbolic toral automorphism with matrix [[2,1],[1,1]] of determinant 1. Ergodic, mixing, Bernoulli, with positive Lyapunov exponent…
Lyapunov exponent
Exponential rate of separation of nearby trajectories. Positive top Lyapunov exponent is the hallmark of chaos. For smooth ergodic systems,…
Oseledec multiplicative ergodic theorem
For an ergodic measure-preserving system T with log-integrable cocycle A: X → GL(d, ℝ), almost every x has a Lyapunov flag V₁(x) ⊃ ⋯ ⊃…
Topological entropy
Exponential growth rate of (n, ε)-separated orbit segments. For a full k-shift h_top = log k. Invariant under topological conjugacy and an…
Variational principle (entropy)
For every continuous T on a compact space, the topological entropy equals the supremum of measure-theoretic entropies over T-invariant…
Furstenberg multiple recurrence theorem
For any T measure-preserving, K ≥ 1, and A with μ(A) > 0, there is n with μ(A ∩ T^{-n}A ∩ ⋯ ∩ T^{-(K-1)n}A) > 0. Furstenberg's…
Khinchin recurrence theorem
For a measure-preserving transformation T of a probability space and a set A of positive measure, the set of return times n with μ(A ∩…
Kac return-time lemma
For an ergodic measure-preserving T on (X, μ) and a set A with μ(A) > 0, the first-return time τ_A(x) = inf{n ≥ 1 : T^n x ∈ A} satisfies…
Szemerédi theorem (ergodic form)
Every set of integers with positive upper density contains arbitrarily long arithmetic progressions. Szemerédi 1975 (combinatorial proof).…
Shannon–McMillan–Breiman theorem
For an ergodic measure-preserving T with a finite generating partition P, the normalised information content −(1/n) log μ(P^(n)(x)) of the…
Rohlin tower lemma
Given an aperiodic measure-preserving T on a standard probability space, for every n ∈ ℕ and ε > 0 there exists a base set B such that B,…
Pesin entropy formula
For a C^{1+α} diffeomorphism f of a compact manifold preserving a smooth (SRB) measure μ, the Kolmogorov–Sinai entropy equals the…
Pinsker algebra
The Pinsker σ-algebra Π(T) of a measure-preserving system is the largest sub-σ-algebra on which T has zero entropy; equivalently the…
Parry measure (maximum entropy)
On an irreducible subshift of finite type Σ_A, the measure of maximal entropy is unique and, writing λ for the spectral radius of A with…