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AuthorRybakowski, Krzysztof P. author
TitleThe Homotopy Index and Partial Differential Equations [electronic resource] / by Krzysztof P. Rybakowski
ImprintBerlin, Heidelberg : Springer Berlin Heidelberg, 1987
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Descript XII, 208 p. online resource


The homotopy index theory was developed by Charles Conley for twoยญ sided flows on compact spaces. The homotopy or Conley index, which provides an algebraic-topologiยญ cal measure of an isolated invariant set, is defined to be the hoยญ 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 inยญ dex pair and is invariant under homotopic perturbations of the flow. Moreover, the homotopy index generalizes the Morse index of a nQndeยญ generate critical point p with respect to a gradient flow on a comยญ pact manifold. In fact if the Morse index of p is k, then the homoยญ 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 and quasi-index pairs -- 1.8 Some special maps used in the construction of the Morse index -- 1.9 The Categorial Morse index -- 1.10 The homotopy index and its basic properties -- 1.11 Linear semiflows. Irreducibility -- 1.12 Continuation of the homotopy index -- II Applications to partial differential equations -- 2.1 Sectorial operators generated by partial differential operators -- 2.2 Center manifolds and their approximation -- 2.3 The index product formula -- 2.4 A one-dimensional example -- 2.5 Asymptotically linear systems -- 2.6 Estimates at zero and nontrivial solution of elliptic equations -- 2.7 Positive heteroclinic orbits of second-order parabolic equations -- 2.8 A homotopy index continuation method and periodic solutions of second-order gradient systems -- III Selected topics -- 3.1 Repeller-attractor pairs and Morse decompositions -- 3.2 Block pairs and index triples -- 3.3 A Morse equation -- 3.4 The homotopy index and Morse theory on Hilbert manifolds -- 3.5 Continuation of the categorial Morse index along paths -- Bibliographical notes and comments

Mathematics Mathematical analysis Analysis (Mathematics) Manifolds (Mathematics) Complex manifolds Mathematics Manifolds and Cell Complexes (incl. Diff.Topology) Analysis


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