Suspension (dynamical systems)

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

$X_{r}=\{(x,t):0\leq t\leq r(x),x\in X\}/(x,r(x))\sim (f(x),0).$

The suspension of $(X,f)$ with roof function $r$ is the semiflow $f_{t}:X_{r}\to X_{r}$ induced by the time translation $T_{t}:X\times \mathbb {R} \to X\times \mathbb {R} ,(x,s)\mapsto (x,s+t)$.

If $r(x)\equiv 1$, then the quotient space is also called the mapping torus of $(X,f)$.