weak mixing

弱混合

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

Other names / variants: measure-theoretic weak mixing

Translation evidence: Literature usage

Meaning in context

A probability-preserving map is weakly mixing if $N^{-1}\sum_{n=0}^{N-1}|\mu(A\cap T^{-n}B)-\mu(A)\mu(B)|\to0$ for every measurable $A,B$.

保概率变换的任意两个固定事件，其相关概率误差的绝对值在时间上的平均趋于0。

Example

A Bernoulli shift is weakly mixing, while an irrational circle rotation is ergodic but not weakly mixing.

Usage and distinctions

The absolute values prevent cancellation; removing them gives an ergodicity criterion rather than weak mixing.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Henk Bruin — Notes on Ergodic TheorySection 12, Definition 14 and Theorem 16 · 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，第二章 混合性，第1节 强混合、弱混合 · Checked 2026-10-14

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