Suspension (dynamical systems)

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

The suspension of ( X , f ) with roof function r is the semiflow f t : X r → X r induced by the time-translation T t : X × R → X × R , ( x , s ) ↦ ( x , s + t ) .

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