Continuation maps in Morse theory

Abstract

Dynamical processes in nature may be described by differential equations. These differential equations depend on parameters that are not known precisely. One only captures qualities of differential equations that remain visible under small perturbations of the equation. Mathematically this means that one is interested in geometric invariants of dynamical systems. Non-degenerate critical points of the dynamical system are such examples. They stay non-degenerated while degenerated critical points may be perturbed away or produce two non-degenerated critical points. So degeneracies can be perturbed away. That is true if one considers a single differential equation. The situation changes if one considers a family of differential equations. In a one parameter family of dynamical systems it might well be that the degeneracies can’t be perturbed away. In this thesis we consider generic one parameter families of gradients and three different continuation maps determined by a given family. We prove that all continuation maps are equal. More precisely: Let f : M → R be a Morse function on a not necessarily compact manifold M and g a metric on M such that the flow φ induced by ẋ = −∇f(x) is Morse-Smale on a compact isolated invariant set S. The Morse homology (with integer coefficients) defined by the Morse-Smale triple (S, f, g) is denoted by HM∗(S, f, g) and its homological Conley index is denoted by HC∗(S, f, g). A continuation (S, f, g) := {(Sλ, fλ, gλ)}06λ61 connecting two Morse-Smale triples (S0, f0, g0) and (S1, f1, g1) determines three different continuation maps (1) Φcon(S, f, g) : HC∗(S0, f0, g0) → HC∗(S1, f1, g1) (2) Φflo(S, f, g) : HM∗(S0, f0, g0) → HM∗(S1, f1, g1) (3) Φbif (S, f, g) : HM∗(S0, f0, g0) → HM∗(S1, f1, g1). The map Φcon(S, f, g) is the Conley continuation map [1]. The map Φflo(S, f, g) is the Floer continuation map defined by counting orbits of ẋ = −∇ft(x), where λ is replaced by the time parameter t. The map Φbif (S, f, g) is the Floer bifurcational continuation map and is defined by studying the change of the orbit structure of ẋ = −∇λfλ(x) at bifurcations. The Conley index of a Morse-Smale triple is isomorphic to its Morse homology α : HC∗(S, f, g) → HM∗(S, f, g) see Theorem A. In Theorem C we prove that the continuation map Φflo(S, f, g) is isomorphic to the Conley continuation map Φcon(S, f, g) meaning Φcon(S, f, g) = α ◦ Φflo(S, f, g) ◦ α . If f = {fλ}λ∈[0,1] is a family of Morse functions fλ we prove in Theorem D that Φcon(S, f, g) = α ◦ Φbif (S, f, g) ◦ α . We think that the assumption that fλ is Morse for all λ ∈ [0, 1] can be removed.