Let H be a real infinite dimensional and separable Hilbert space. With an isolated invariant set inv(N) of a flow φ t generated by an LS-vector field f : H ⊇ Ω → H, f (x) = Lx + K(x), where L: H → H is strongly indefinite linear operator and K: H ⊇ Ω → H is completely continuous, one can associate a homotopy invariant h LS (inv(N), φ t) called the LS-Conley index. In fact, this is a homotopy type of a finite CWcomplex. We define the Betti numbers and hence the Euler characteristic of such index and prove the formula relating these numbers to the Leray-Schauder degree deg LS (b f , N, 0), where b f : H ⊇ Ω → H is defined as b f (x) = x + L −1 K(x).
Abstract. In this paper we introduce a new compactness condition -Property-(C)-for flows in (not necessary locally compact) metric spaces. For such flows a Conley type theory can be developed. For example (regular) index pairs always exist for Property-(C) flows and a Conley index can be defined. An important class of flows satisfying this compactness condition are LS-flows. We apply E-cohomology to index pairs of LS-flows and obtain the E-cohomological Conley index. We formulate a continuation principle for the E-cohomological Conley index and show that all LS-flows can be continued to LS-gradient flows. We show that the Morse homology of LS-gradient flows computes the E-cohomological Conley index. We use Lyapunov functions to define the Morse-Conley-Floer cohomology in this context, and show that it is also isomorphic to the E-cohomological Conley index.
The paper is concerned with the Morse equation for flows in a representation of a compact Lie group. As a consequence of this equation we give a relationship between the equivariant Conley index of an isolated invariant set of the flow given by˙ = − ∇ ( ) and the gradient equivariant degree of ∇ . Some multiplicity results are also presented.
MSC:37J35, 57R70, 47H11
Abstract.A special case of G-equivariant degree is defined, where G = Z 2 , and the action is determined by an involution T :The presented construction is self-contained. It is also shown that two T -equivariant gradient maps f, g : (R n , S n−1 ) → (R n , R n \ {0}) are T -homotopic iff they are gradient T -homotopic. This is an equivariant generalization of the result due to Parusiński.
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