Nested sequences of index filtrations and continuation of the connection matrix

AbstractIn this paper, we prove the existence of nested sequences of index filtrations for convergent sequences of (admissible) semiflows on a metric space. This result is new even in the context of flows on a locally compact space. The nested index filtration theorem implies the continuation of homology index braids which, in turn, implies the continuation of connection matrices in the infinite-dimensional Conley index theory

On curved squeezing and Conley index

We consider reaction-diffusion equations on a family of domains depending on a parameter \eps> 0. As \eps\to 0, the domains degenerate to a lower dimensional manifold. Using some abstract results introduced in the recent paper \cite{\rfa{CR2}} we show that there is a limit equation as \eps\to 0 and obtain various convergence and admissibility results for the corresponding semiflows. As a consequence, we also establish singular Conley index and homology index continuation results. Under an additional dissipativeness assumption, we also prove existence and upper-semicontinuity of global attractors. The results of this paper extend and refine earlier results of \cite{\rfa{CR1}} and \cite{\rfa{PRR}}

Curved squeezing of unbounded domains and tail estimates

Using a resolvent convergence result from [7] we prove Conley index and index braid continuation results for reaction-diffusion equations on singularly perturbed unbounded curved squeezed domains

Some recent results in the homotopy index theory in infinite dimensions

Si fornisce un criterio di ammissibilità nell’ambito della teoria dell'indice di omotopìa in spazi metrici e si confronta la condizione di ammissibilità con la condizione di Palais-Smale. Nel caso di problemi variazionali, si collega l’indice di omotopìa alla nozione di gruppi critici di un punto critico. Infine, si applica la teoria dell’indice di omotopìa per stabilire un «principio di perequazione» per soluzioni periodiche di sistemi del secondo ordine di tipo gradiente.In this note we give a criterion for admissibility in the homotopy index theory on metric spaces and compare admissibility with the Palais-Smale condition. For variational problems, we relate the homotopy index to the concept of critical groups of a critical point. Finally, we use the homotopy index to establish an «averaging principle» for periodic solutions of second order gradient systems

The suspension isomorphism for cohomology index braids

Let $X$ be a metric space, $\pi$ be a local semiflow on $X$, $k\in{\mathbb N}$, $E$ be a $k$-dimensional normed real vector space and $\widetilde\pi$ be the semiflow generated by the equation $\dot y=Ly$, where L\co E\to E is a linear map whose all eigenvalues have positive real parts. We show in this paper that for every admissible isolated $\pi$-invariant set $S$ there is a well-defined isomorphism of degree $k$ from the (Alexander-Spanier)-cohomology categorial Conley-Morse index of $(\pi,S)$ to the cohomology categorial Conley-Morse index of $(\pi\times\widetilde\pi,S\times\{0\})$ such that the family of these isomorphisms commutes with cohomology index sequences. This extends previous results by Carbinatto and Rybakowski to the Alexander-Spanier-cohomology case

On critical groups and the homotopy index in Morse theory on Hilbert manifolds

Sia U un aperto nello spazio di Hilbert H, $\varphi\epsilon C^{2-}(U,\mathbf{R)\textrm{,}}\xi\epsilon U$ un punto critico isolato di $\varphi$, e $\pi$il flusso generato dalle soluzioni di $\dot{u}$=-$\triangle\varphi(u)$. Se $\xi$ ha un intorno fortemente ammissibile, allora i gruppi critici di ($\varphi$, $\xi$) nel senso di Rothe sono isomorfi ai gruppi di omologia dell'indice di omotopia di ($\pi,\left\{ \xi\right\} )$ (Teorema 2). Se $\varphi\epsilon C^{2}(U,\mathbf{R})$, $\varphi''(\xi)$ è un'applicazione di Fredholm, ma $\xi$ non ha un intorno fortemente ammissibile, allora tutti i gruppi critici di ($\varphi,\xi)$ sono uguali a zero (banali) (Teorema 4).Let U be open in the Hilbert space H, $\varphi\epsilon C^{2-}(U,\mathbf{R)\textrm{,}}\xi\epsilon U$ be an isolated criticai point of $\varphi$, and $\pi$ be the flow generated by the solutions of $\dot{u}$=-$\triangle\varphi(u)$. If $\xi$ has a strongly admissible neighborhood, then the critical groups of ($\varphi$, $\xi$) are isomorphic to the homology groups of the homotopy index of ($\pi,\left\{ \xi\right\} )$ (Theorem 2). If $\varphi\epsilon C^{2}(U,\mathbf{R})$, $\varphi''(\xi)$ is a Fredholm operator, but $\xi$ does not have a strongly admissible neighborhood then all critical groups of ($\varphi,\xi)$ are trivial (Theorem 4)

CONLEY INDEX CONTINUATION FOR SINGULARLY PERTURBED HYPERBOLIC EQUATIONS

in gratitude Abstract. Let Ω ⊂ R N, N ≤ 3, be a bounded domain with smooth boundary, γ ∈ L 2 (Ω) be arbitrary and φ: R → R be a C 1-function satisfying a subcritical growth condition. For every ε ∈]0, ∞ [ consider the semiflow πε on H 1 0 (Ω) × L2 (Ω) generated by the damped wave equation ε∂ttu + ∂tu = ∆u + φ(u) + γ(x) x ∈ Ω, t> 0, u(x, t) = 0 x ∈ ∂Ω, t> 0. Moreover, let π ′ be the semiflow on H1 0 (Ω) generated by the parabolic equation ∂tu = ∆u + φ(u) + γ(x) x ∈ Ω, t> 0, u(x, t) = 0 x ∈ ∂Ω, t> 0. Let Γ: H2 (Ω) → H1 0 (Ω) × L2 (Ω) be the imbedding u ↦ → (u, ∆u + φ(u) + γ). We prove in this paper that every compact isolated π ′-invariant set K ′ lies in H2 (Ω) and the imbedded set K0 = Γ(K ′) continues to a family Kε, ε ≥ 0 small, of isolated πε-invariant sets having the same Conley index as K ′. This family is upper-semicontinuous at ε = 0. Moreover, any (partially ordered) Morse-decomposition of K ′, imbedded into H1 0 (Ω) × L2 (Ω) via Γ, continues to a family of Morse decompositions of Kε, for ε ≥ 0 small. This family is again upper-semicontinuous at ε = 0. These results extend and refine some upper semicontinuity results for attractors obtained previously by Hale and Raugel