Suspension (dynamical systems)

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

:$X\_r=\backslash \{(x,t):0\backslash le\; t\backslash le\; r(x),x\backslash in\; X\backslash \}/(x,r(x))\backslash sim(f(x),0).$

The suspension of $(X,f)$ with roof function $r$ is the semiflowM. Brin and G. Stuck, Introduction to Dynamical Systems, Cambridge University Press, 2002. $f\_t:X\_r\backslash to\; X\_r$ induced by the time translation $T\_t:X\backslash times\backslash mathbb\{R\}\backslash to\; X\backslash times\backslash mathbb\{R\},\; (x,s)\backslash mapsto\; (x,s+t)$.

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