The discrete Conley index for non-invariant sets and detection of chaos

The relationship between dynamical systems and topology dates from the work of Poincare. Since then, algebraic topology has provided significant techniques for the investigation of non-linear systems. An important contribution to this area has been made by Charles Conley, whose index is a topological generalization of the Morse index.;The first chapter of this dissertation briefly surveys the development of the Conley index theory from its origins to the more recently introduced discrete Conley index and its latest applications in detecting periodic orbits and chaos.;In the second chapter we present a cohomological Conley index for discrete-time dynamical systems in non-locally compact spaces, similar to the one constructed by Degiovanni and Mrozek. This is a discrete version of the continuous Conley index introduced by Benci and generalizes similar indices developed by Mrozek and Rybakowski. A few applications in studying hyperbolic and pseudo-hyperbolic operators are mentioned. However, the main purpose of this chapter is to produce the tools and techniques for the next chapter.;The third chapter constitutes the core of this dissertation. We define a new cohomological Conley index associated to some isolating neighborhood sequences, extending previous constructions of Easton, Srzednicki and of the author. Our construction uses a very general Leray functor on the category of graded directed systems of modules and homomorphisms, extending a similar construction of Mrozek. This index satisfies the fundamental Wazewski property, summation property and continuation property, common to all Conley index type of invariants. Under special conditions, it turns out to be an index of non-invariant sets, related to the Conley index for decompositions of isolated invariant sets considered by Szymczak. We apply our index to detect periodic orbits, extending results of Mrozek and Srzednicki and to detect chaos through a semi-conjugacy to a shift space, extending results of Mischaikow and Mrozek, Mischaikow, Kwapisz and Carbinatto, Srzednicki, Szymczak. We illustrate our methods on several types of horseshoes. In the end, we provide a unified Conley index theory approach to describe the chaotic behavior of some Julia sets.