We build the Conley Index from the concept of a index pair.
Given an isolated invariant set in a flow , an index pair for is a pair of compact sets , with , satisfying
- and is a neighborhood of ;
- For all and , ;
- For all and , such that .
Conley shows that every isolating invariant set admits an index pair. For an isolated invariant set , we choose some index pair of and the we define, then, the homotopy Conley index of as
- ,
the homotopy type of the quotient space , seen as a topological pointed space.
Analogously, the (co)homology Conley index of is the chain complex
- .
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.
- 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
- M. R. Razvan, On Conley’s fundamental theorem of dynamical systems, 2002.