Suspension (dynamical systems)

Suspension is a construction passing from a map to a flow. Namely, let [math]\displaystyle{ X }[/math] be a metric space, [math]\displaystyle{ f:X\to X }[/math] be a continuous map and [math]\displaystyle{ r:X\to\mathbb{R}^+ }[/math] be a function (roof function or ceiling function) bounded away from 0. Consider the quotient space:

[math]\displaystyle{ X_r=\{(x,t):0\le t\le r(x),x\in X\}/(x,r(x))\sim(f(x),0). }[/math]

The suspension of [math]\displaystyle{ (X,f) }[/math] with roof function [math]\displaystyle{ r }[/math] is the semiflow [math]\displaystyle{ f_t:X_r\to X_r }[/math] induced by the time translation [math]\displaystyle{ T_t:X\times\mathbb{R}\to X\times\mathbb{R}, (x,s)\mapsto (x,s+t) }[/math].

If [math]\displaystyle{ r(x)\equiv 1 }[/math], then the quotient space is also called the mapping torus of [math]\displaystyle{ (X,f) }[/math].