suspension flow construction

扭扩

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

Other names / variants: suspension

Translation evidence: Literature usage

Meaning in context

For a homeomorphism $T:X\to X$ of a nonempty compact metric space and a continuous roof $r:X\to(0,\infty)$, the global real-time flow on the quotient of $\{(x,s):0\le s\le r(x)\}$ identifying $(x,r(x))$ with $(Tx,0)$, obtained by advancing the height coordinate.

对非空紧致度量空间 $X$ 上的同胚 $T:X\to X$ 及连续正顶函数 $r:X\to(0,\infty)$，将 $\{(x,s):0\le s\le r(x)\}$ 中各柱顶 $(x,r(x))$ 与下一柱底 $(Tx,0)$ 粘合，并沿高度前进所得到的全局实数时间流。

Example

Suspending the identity on a single point with roof $1$ gives unit-speed rotation of a circle.

Usage and distinctions

Compactness and a positive continuous roof give a positive minimum, preventing infinitely many roof crossings in finite time. Sarig's cited construction is a measurable suspension semiflow variant; this record fixes the topological compact-base flow convention. Noninvertible bases lead to semiflows. The inspected Chinese source calls suspension 扭扩, a documented naming variant.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Omri Sarig — Lecture Notes on Ergodic Theory§1.6.5, Definition 1.20 and preceding assumption inf r > 0, printed p.30 / PDF p.38 (measurable suspension semiflow construction; this record uses the scoped topological variant with compact metric base, homeomorphism and continuous positive roof) · Checked 2026-10-14

Translation sources

1. 陈绥阳、褚蕾蕾 — 动力系统基础及其方法（科学出版社预览）§2.1，注2.1.1，p.11：扭扩（suspension） · Checked 2026-10-14

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