Suspension (dynamical systems) explained

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 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^[1] f _{t:X} _{r\to} X _{r} induced by the time translation T _{t:X x R\to} X x 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) .

Notes and References

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