The Homotopy Index and Partial Differential Equations [Paperback]

The homotopy index theory was developed by Charles Conley for two? sided flows on compact spaces. The homotopy or Conley index, which provides an algebraic-topologi? cal measure of an isolated invariant set, is defined to be the ho? motopy type of the quotient space N /N , where is a certain 1 2 1 2 compact pair, called an index pair. Roughly speaking, N1 isolates the invariant set and N2 is the exit ramp of N . 1 It is shown that the index is independent of the choice of the in? dex pair and is invariant under homotopic perturbations of the flow. Moreover, the homotopy index generalizes the Morse index of a nQnde? generate critical point p with respect to a gradient flow on a com? pact manifold. In fact if the Morse index of p is k, then the homo? topy index of the invariant set {p} is Ik - the homotopy type of the pointed k-dimensional unit sphere.The homotopy index theory was developed by Charles Conley for two? sided flows on compact spaces. The homotopy or Conley index, which provides an algebraic-topologi? cal measure of an isolated invariant set, is defined to be the ho? motopy type of the quotient space N /N , where is a certain 1 2 1 2 compact pair, called an index pair. Roughly speaking, N1 isolates the invariant set and N2 is the exit ramp of N . 1 It is shown that the index is independent of the choice of the in? dex pair and is invariant under homotopic perturbations of the flow. Moreover, the homotopy index generalizes the Morse index of a nQnde? generate critical point p with respect to a gradient flow on a com? pact manifold. In fact if the Morse index of p is k, then the homo? topy index of the invariant set {p} is Ik - the homotopy type of the pointed k-dimensional unit sphere.I The homotopy index theory.- 1.1 Local semiflows.- 1.2 The no blow-up condition. Convergence of semiflows.- 1.3 Isolated invariant sets and isolating blocks.- 1.4 Admissibility.- 1.5 Existence of isolating blocks.- 1.6 Homotopies and inclusion induced maps.- 1.7 Index lc»