Conley index theory

A key role in the theory is played by the notions of isolating neighborhood ${\displaystyle N}$ and isolated invariant set ${\displaystyle S}$. The Conley index ${\displaystyle h(S)}$ is the homotopy type of a space built from a certain pair ${\displaystyle (N_{1},N_{2})}$ of compact sets called an index pair for ${\displaystyle S}$. Charles Conley showed that index pairs exist and that the index of ${\displaystyle S}$ is independent of the choice of the index pair. In the special case of the negative gradient flow of a smooth function, the Conley index of a nondegenerate (Morse) critical point of index ${\displaystyle N}$ is the pointed homotopy type of the k-sphere S^k.

A deep theorem due to Conley asserts continuation invariance: Conley index is invariant under certain deformations of the dynamical system. Computation of the index can, therefore, be reduced to the case of the diffeomorphism or a vector field whose invariant sets are well understood.

If the index is nontrivial then the invariant set S is nonempty. This principle can be amplified to establish existence of fixed points and periodic orbits inside N.

We build the Conley Index from the concept of a index pair.

Given an isolated invariant set ${\displaystyle S}$ in a flow ${\displaystyle \phi }$, an index pair for ${\displaystyle S}$ is a pair of compact sets ${\displaystyle (N_{1},N_{2})}$, with ${\displaystyle N_{2}\subset N_{1}}$, satisfying

• ${\displaystyle S={\text{Inv}}(N_{1}/N_{2})}$ and ${\displaystyle N_{1}/N_{2}}$ is a neighborhood of ${\displaystyle S}$;
• For all ${\displaystyle x\in N_{2}}$ and ${\displaystyle t>0}$, ${\displaystyle \phi ([0,t],x)\subset N_{1}\Rightarrow \phi ([0,t],x)\subset N_{2}}$;
• For all ${\displaystyle x\in N_{1}}$ and ${\displaystyle t>0}$, ${\displaystyle \phi (t,x)\not \in N_{1}\Rightarrow \exists t'\in [0,t]}$ such that ${\displaystyle \phi (t',x)\in N_{2}}$.

Conley shows that every isolating invariant set admits an index pair. For an isolated invariant set ${\displaystyle S}$, we choose some index pair ${\displaystyle (N_{1},N_{2})}$ of ${\displaystyle S}$ and the we define, then, the homotopy Conley index of ${\displaystyle S}$ as

${\displaystyle h(S,\phi ):=[(N_{1}/N_{2},[N_{2}])]}$,

the homotopy type of the quotient space ${\displaystyle (N_{1}/N_{2},[N_{2}])}$, seen as a topological pointed space.

Analogously, the (co)homology Conley index of ${\displaystyle S}$ is the chain complex

${\displaystyle CH_{\bullet }(S,\phi )=H_{\bullet }(N_{1}/N_{2},[N_{2}])}$.

We remark that also Conley showed that the Conley index is independent of the choice of an index pair, so that the index is well defined.

Some of the most important properties of the index are direct consequences of its definition, inheriting properties from homology and homotopy. Some of them include the following:

• If ${\displaystyle h(S)\neq 0}$, then ${\displaystyle S\neq \emptyset }$;

• If ${\displaystyle S=\cup _{i=1}^{n}M_{i}}$, where each ${\displaystyle M_{i}}$ is an isolated invariant set, then ${\displaystyle CH_{k}(S)=\oplus _{i=1}^{n}CH_{k}(M_{i})}$;

• The Conley index is homotopy invariant.

Notice that, a Morse set is an isolated invariant set, so that the conley index is defined for it.

• Charles Conley, Isolated invariant sets and the Morse index. CBMS Regional Conference Series in Mathematics, 38. American Mathematical Society, Providence, R.I., 1978 ISBN 0-8218-1688-8
• Thomas Bartsch (2001) [1994], "Conley index", Encyclopedia of Mathematics, EMS Press
• John Franks, Michal Misiurewicz, Topological methods in dynamics. Chapter 7 in Handbook of Dynamical Systems, vol 1, part 1, pp 547–598, Elsevier 2002 ISBN 978-0-444-82669-5
• Jürgen Jost, Dynamical systems. Examples of complex behaviour. Universitext. Springer-Verlag, Berlin, 2005 ISBN 978-3-540-22908-7
• Konstantin Mischaikow, Marian Mrozek, Conley index. Chapter 9 in Handbook of Dynamical Systems, vol 2, pp 393–460, Elsevier 2002 ISBN 978-0-444-50168-4
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• Separation of Topological Singularities (Wolfram Demonstrations Project)

• Conley's fundamental theorem of dynamical systems