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Construction and properties of the Conley index

100%

Mrozek M.

tom 47

nr 1

29-40

Shape index and other indices of Conley type for local maps on locally compact Hausdorff spaces

100%

Mrozek M.

Fundamenta Mathematicae

1994

tom 145

nr 1

15-37

We present a scheme for constructing various Conley indices for locally defined maps. In particular, we extend the shape index of Robbin and Salamon to the case of a locally defined map in a locally compact Hausdorff space. We compare the shape index with the cohomological Conley index for maps. We also prove the commutativity property of the Conley index, which is analogous to the commutativity property of the fixed point index.

Homology of representable sets

63%

Mrozek M. , Batko B.

tom 97

nr 3

243-252

We generalize the notion of cubical homology to the class of locally compact representable sets in order to propose a new convenient method of reducing the complexity of a set while computing its homology.

Set arithmetic and the enclosing problem in dynamics

63%

Mrozek M. , Zgliczyński P.

2000

tom 74

nr 1

237-259

We study the enclosing problem for discrete and continuous dynamical systems in the context of computer assisted proofs. We review and compare the existing methods and emphasize the importance of developing a suitable set arithmetic for efficient algorithms solving the enclosing problem.

On a generalization of the Conley index

51%

Mrozek M. , Reineck J. F. , Srzednicki R.

tom 47

nr 1

157-171

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 some of the information which is lost on collapse. As an example, is it possible that two repelling periodic orbits which are not homotopic in the punctured plane are related by continuation? Clearly one cannot be continued to the other as periodic orbits, but the index of such a periodic orbit is the same as the index of the disjoint union of two rest points, so the question of continuation as isolated invariant sets is far less obvious.

Index pairs and the fixed point index for semidynamical systems with discrete time

50%

Mrozek M.

Fundamenta Mathematicae

1989

tom 133

nr 3

179-194

Foreword, Contents

32%

Mischaikow K. , Mrozek M. , Zgliczyński P.

tom 47

nr 1

1-7