An approach to the shape Conley index without index pairs

Abstract

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.

On the shape Conley index theory of semiflows on complete metric spaces

“…Second, it allows to calculate shape indices and Morse equations of invariant sets by using only the unstable manifolds of the invariant sets and their Morse sets. Robbin and Salamon's approach to the shape index theory was further developed in the works of Mrozek [12] and Sánchez-Gabites [15] for dynamical systems on locally compact spaces.…”

Čech cohomology, homoclinic trajectories and robustness of non-saddle sets

“…. , γ c independent non-torsion cohomology classes in im i * n−1 and the result follows.A direct consequence of Theorem 3.6 is the following result that generalizes[24, Theorem 4.6].…”

“…In particular, we see that phase spaces with little cohomology do not support isolated non-saddle sets with overly complicated regions influence. This is motivated by some results from [23,24] about certain unstable attractors. In addition, motivated by the results in [6] and [3] about the continuation properties of isolated non-saddle sets we find necessary and sufficient conditions for the property of being non-saddle to be robust for families of smooth flows defined on smooth manifolds without further assumptions about the dimension or cohomology of the phase space.…”

“…Dynamical complexity of the region of influence of isolated non-saddle sets and the cohomology of the phase space. In this section we study to what extent the topology of the phase space and the way in which the isolated non-saddle continuum K sits in it at the cohomological level affects the structure of the region of influence of K. Some results in this spirit were obtained by Sánchez-Gabites [23,24] in the case of isolated attractors without external explosions and by Barge and Sanjurjo [6] for isolated non-saddle continua in closed manifolds. Although the results we present here can be regarded as generalizations of the aforementioned results, they stress again that the structure of the region of influence of an isolated non-saddle set is much more subtle than the region of attraction of an isolated attractor without external explosions.…”

Conley index theory without index pairs. I: the point-set level theory

“…Remark 1.1. There is another approach to developing Conley index theory without using index pairs by Sánchez-Gabites [Sán11].…”