Poincaréåterkomst

Poincaré återkomst, commonly known in English as the Poincaré recurrence theorem, is a result in dynamical systems and ergodic theory named after Jules Henri Poincaré, who proved it in 1890. The theorem concerns the behavior of measure-preserving dynamical systems with finite total measure. Roughly speaking, it states that any region of the state space with positive measure will be revisited by the system's trajectory arbitrarily many times.

Formal statement: Let (X, Σ, μ) be a measure space with μ(X) < ∞ and T: X → X a measurable transformation that preserves μ (μ(T^{-1}(B)) = μ(B) for all B ∈ Σ). If A ⊆ X has μ(A) > 0, then for μ-almost every x ∈ A there exist infinitely many integers n ≥ 1 such that T^n x ∈ A. In words, almost every point in A returns to A infinitely often under iterations of T.

The theorem holds for discrete-time systems and has analogues for continuous-time flows. It implies recurrence: in any finite-measure, measure-preserving system, almost every starting state in any positive-measure region will re-enter that region repeatedly over time. It does not, however, guarantee periodicity or a regular return schedule; returns can be irregular and far apart.

Consequences and context: Poincaré recurrence is a foundational result in ergodic theory and underpins arguments about how finite systems explore their state space over time. It contributes to the intuition behind statistical mechanics and the long-term behavior of dynamical systems. Related concepts include ergodicity, Kac's lemma, and various recurrence theorems that refine or extend the basic idea.