Poincaré recurrence theorem

庞加莱回复定理

Dynamical Systems · Graduate · Mathematics Resources editorial compilation (AI-assisted)

Other names / variants: Poincaré recurrence

Translation evidence: Literature usage

Meaning in context

For a probability-preserving transformation $T$ and measurable $A$, almost every $x\in A$ returns to $A$ infinitely often under positive iterates.

保概率系统中，可测集合内几乎所有点在正向迭代下无穷多次返回该集合的定理。

Example

A rotation preserving circle length has almost-everywhere returns to any measurable arc.

Usage and distinctions

Finite total measure is essential for this conclusion; translation on the integers with counting measure can have no returns.

Read the encyclopedia explanation

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Omri Sarig — Lecture Notes on Ergodic Theory§1.2, Theorem 1.1 (Poincaré's Recurrence Theorem) · Checked 2026-10-14

Translation sources

1. 何伟鲲 — 遍历论课程大纲（中国科学院大学）大纲内容，第一章，第2节 回复性、庞加莱回复定理 · Checked 2026-10-14

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