Suspension (dynamical systems)

Suspension is a construction passing from a map to a flow. Namely, let  X {\displaystyle X} be a metric space,  f : X -> X {\displaystyle f:X\to X} be a continuous map and  r : X -> R + {\displaystyle 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 ≤ t ≤ r ( x ) , x ∈ X } / ( x , r ( x ) ) ∼ ( f ( x ) , 0 ) . {\displaystyle 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 ) {\displaystyle (X,f)} with roof function  r {\displaystyle r} is the semiflow[1] f t : X r -> X r {\displaystyle f_{t}:X_{r}\to X_{r}} induced by the time translation T t : X × R -> X × R , ( x , s ) ↦ ( x , s + t ) {\displaystyle T_{t}:X\times \mathbb {R} \to X\times \mathbb {R} ,(x,s)\mapsto (x,s+t)} .

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

## References

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1. ^ M. Brin and G. Stuck, Introduction to Dynamical Systems, Cambridge University Press, 2002.