Homotopy invariance of the Conley index and local Morse homology in Hilbert spaces

Abstract

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.

Connected components of the space of proper gradient vector fields

“…The proof of Proposition 2.4 remain unchanged in this case. Since the E-cohomological Conley index is invariant under homotopies (Theorem 2.12 in [3]), the conclusion of this note is also true for Hilbert spaces.…”

“…Note first that instead of using Morse cohomology groups one can use Conley index theory instead. If instead of R n we take a separable Hilbert space and assume that the vector fields are compact perturbations of a fixed bounded self-adjoint Fredholm operator then the cohomological invariant, called Ecohomological Conley index, is still well defined (see [3,9] or [10]). The proof of Proposition 2.4 remain unchanged in this case.…”

The existence of isolating blocks for multivalued semiflows

“…Since u ε n ∈ D ε 0 u n 0 for all n, by Lemma 5.4 we obtain that up to a subsequence u ε n → u ∈ D ε 0 (u 0 ) uniformly on bounded sets of [0, +∞). Hence, u (•) is a strong solution to problem (14) with h ∈ L 2 loc (0, +∞; L 2 (Ω)), h (t) ∈ F ε 0 (u(t)) for a.a. t, where F ε 0 is the map (11) for f ε 0 .…”

“…The importance of this topology definition can be measured by the large amount of studies that followed the above references. Just to cite a few of them, this concept was used in applications (see for instance, [18,15]), it was also defined for flows on Hilbert spaces (see [10,4,11]), for non-autonomous semiflows on Banach spaces [12] and also for multivalued semiflows (see [8,16]).…”