An approach to the shape Conley index without index pairs

作者：Sanchez Gabites J J^*

来源：Revista Matematica Complutense, 2011, 24(1): 95-114.

DOI：10.1007/s13163-010-0031-x

• Dynamical systems
• Conley index
• Intrinsic topology
• Morse equations
• Unstable manifold
• Instability depth

Robbin and Salamon proved that the shape of the homotopy Conley index of an isolated invariant set K coincides with the shape of the one-point compactification of the unstable region W (u) (K) of K endowed with a certain topology which they called the intrinsic topology. In this paper an equivalent, simplified definition of the latter is given in elementary terms, without resorting to index pairs whatsoever. We then show how our approach allows for a development of the shape index and its basic properties in an index pair-free fashion and simplifies some classical constructions, such as that of the Morse equations of a Morse decomposition. As a final application, some embrionary work about the geometry of W (u) (K)/K is presented together with an argument about how this may contribute to our understanding of the relations between how an attractor sits in its basin of attraction and the complexity of the dynamics in the latter.

• 出版日期2011-4