Suspension (dynamical systems)

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

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

The suspension of ${\displaystyle (X,f)}$ with roof function ${\displaystyle r}$ is the semiflow^[1] ${\displaystyle f_{t}:X_{r}\to X_{r}}$ induced by the time translation ${\displaystyle T_{t}:X\times \mathbb {R} \to X\times \mathbb {R} ,(x,s)\mapsto (x,s+t)}$.

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