Displaying similar documents to “Index pairs and the fixed point index for semidynamical systems with discrete time”

The Conley index theory: A brief introduction

Konstantin Mischaikow (1999)

Banach Center Publications

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A brief introduction to the Conley index theory is presented. The emphasis is the fundamental ideas of Conley's approach to dynamical systems and how it avoids some of the difficulties inherent in the study of nonlinear systems.

The Conley index and countable decompositions of invariant sets

Marian Gidea (1999)

Banach Center Publications

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We define a new cohomological index of Conley type associated to any bi-infinite sequence of neighborhoods that satisfies a certain isolation condition. We use this index to study the chaotic dynamics on invariant sets which decompose as countable unions of pairwise disjoint (mod 0) compact pieces.

The Conley index for flows preserving generalized symmetries

Artur Pruszko (1999)

Banach Center Publications

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Topological spaces with generalized symmetries are defined and extensions of the Conley index of a compact isolated invariant set of the flow preserving the structures introduced are proposed. One of the two new indexes is constructed with no additional assumption on the examined set in terms of symmetry invariance.

On a generalization of the Conley index

Marian Mrozek, James Reineck, Roman Srzednicki (1999)

Banach Center Publications

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In this note we present the main ideas of the theory of the Conley index over a base space introduced in the papers [7, 8]. The theory arised as an attempt to solve two questions concerning the classical Conley index. In the definition of the index, the exit set of an isolating neighborhood is collapsed to a point. Some information is lost on this collapse. In particular, topological information about how a set sits in the phase space is lost. The first question addressed is how to retain...

On foundations of the Conley index theory

Roman Srzednicki (1999)

Banach Center Publications

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The Conley index theory was introduced by Charles C. Conley (1933-1984) in [C1] and a major part of the foundations of the theory was developed in Ph. D. theses of his students, see for example [Ch, Ku, Mon]. The Conley index associates the homotopy type of some pointed space to an isolated invariant set of a flow, just as the fixed point index associates an integer number to an isolated set of fixed points of a continuous map. Examples of isolated invariant sets arise naturally in the...

Index filtrations and Morse decompositions for discrete dynamical systems

P. Bartłomiejczyk, Z. Dzedzej (1999)

Annales Polonici Mathematici

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On a Morse decomposition of an isolated invariant set of a homeomorphism (discrete dynamical system) there are partial orderings defined by the homeomorphism. These are called admissible orderings of the Morse decomposition. We prove the existence of index filtrations for admissible total orderings of a Morse decomposition and introduce the connection matrix in this case.