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:


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

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 $$(X,f)$$.