Year 2017

• Homotopy invariance of the Conley index and local Morse homology in Hilbert spaces

  Publication

  • M. Izydorek
  • T. O. Rot
  • M. Starostka
  • M. Styborski
  • R. C. Vandervorst

  - JOURNAL OF DIFFERENTIAL EQUATIONS - Year 2017

  In this paper we introduce a new compactness condition — Property-(C) — for flows in (not necessary locally compact) metric spaces. For such flows a Conley type theory can be developed. For example (regular) index pairs always exist for Property-(C) flows and a Conley index can be defined. An important class of flows satisfying the this compactness condition are LS-flows. We apply E-cohomology to index pairs of LS-flows and obtain...

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Year 2012

• Equivariant Morse equation

  Publication

  • M. Styborski

  - Central European Journal of Mathematics - Year 2012

  The paper is concerned with the Morse equation for flows in a representation of a compact Lie group. As a consequence of this equation we give a relationship between the equivariant Conley index of an isolated invariant set of the flow given by x˙ = − ∇f(x) and the gradient equivariant degree of ∇f. Some multiplicity results are also presented.

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Year 2011

• Morse inequalities via Conley index theory

  Publication

  • M. Izydorek
  • M. Styborski

  - Year 2011

  The relation known as the Morse inequalities can be extended to a more general setting of flows on a locally compact metric spaces (Conley index) as well as dynamical systems on Hilbert spaces (LS-index). This paper is a discourse around this extension. Except the part concerning the LS-index the material is self-contained and has a character of a survey.

Year 2010

• Topological invariants for equivariant flows: Conley index and degree

  Publication

  • M. Styborski

  - Year 2010

  About forty years have passed since Charles Conley defined the homotopy index. Thereby, he generalized the ideas that go back to the calculus of variations work of Marston Morse. Within this long time the Conley index has proved to be a valuable tool in nonlinear analysis and dynamical systems. A significant development of applied methods has been observed. Later, the index theory has evolved to cover such areas as discrete dynamical...

Year 2009

• Conley index in Hilbert spaces and the Leray-Schauder degree

  Publication

  • M. Styborski

  - Topological Methods in Nonlinear Analysis - Year 2009

  Zdefiniowane są liczby Bettiego i charakterystyka Eulera LS-indeksu dla potoków generowanych przez pole zwarte w przestrzeni Hilberta. Główna teza pracy to wzór typu Poincare-Hopfa łączący wspomnianą chatrakterystykę Eulera ze stopniem Leray-Schaudera.

Year 2007

• Degree of T-equivariant maps in R^n

  Publication

  • J. Janczewska
  • M. Styborski

  - BANACH CENTER PUBLICATIONS - Year 2007

  W pracy przedstawiona jest konstrukcja niezmienniczego stopnia topologicznego dla odwzorowań z symetriami działających na przestrzeni euklidesowej z inwolucją. Udowodnione jest twierdzenie, że dwa dopuszczalne i gradientowe odwzorowania niezmiennicze są niezmienniczo homotopijne wtedy i tylko wtedy, gdy są one homotopijne niezmienniczo i gradientowo.

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