topological transitivity

拓扑传递

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

Other names / variants: topologically transitive · 拓扑传递性

Translation evidence: Literature usage

Meaning in context

A continuous self-map $T$ is topologically transitive if for every nonempty open $U,V$ some $n\ge0$ satisfies $T^n(U)\cap V\ne\varnothing$.

连续自映射的任意两个非空开区域，存在某个非负迭代次数把前一区域的一部分送到后一区域。

Example

An irrational circle rotation and the full binary shift are topologically transitive.

Usage and distinctions

A dense forward orbit is equivalent in the usual compact metric setting without isolated points; do not use that equivalence without hypotheses.

Read the encyclopedia explanation

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Gerald Teschl — Ordinary Differential Equations and Dynamical Systems§11.3 On the definition of chaos, open-set definition · Checked 2026-10-14

Translation sources

1. 李健、谭枫 — 关于 Li-Yorke δ-混沌与按序列分布 δ-混沌的等价性Printed p.36，注记3(4)：拓扑传递的开集定义 · Checked 2026-10-14

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