Suspension of a Map

  A discrete map can also be used to generate a continuous flow. A canonical construction for this is the so-called suspension of a map  [1], which is in a sense the inverse of a Poincaré map. Given a discrete map f of an n-dimensional manifold M, it is always possible to construct a flow on an n+1-dimensional manifold formed by the Cartesian product with the original manifold: . This suspension process is illustrated in Figure 4.3(a) where we show a mapping and the flow formed by ``suspending'' this map.

  
Figure 4.3: (a) Construction of a suspension of a map. (b) The suspension is not unique; an arbitrary number of twists can be added.

Each sequence of points of the map becomes an orbit of the flow with the property that if then . The original map is recovered from the suspended flow by a time-T map. The suspension construction is far from unique. For instance, as illustrated in Figure 4.3(b), we could add an arbitrary number of complete twists to this particular suspension and still get an identical time-T map. The number of full twists in the suspended flow is called the global torsion , and it is a topological invariant of the flow independent of the underlying map.