Algorithmic Computation of the Conley Index for Multivalued Maps with No Continuous Selector in Cubical Spaces

Mateusz Przybylski

Abstrakt

In this note we present theory which facilitates the use of Conley index algorithms for cubical multivalued maps constructed from maximal dimensional cubes in the setting when cubes of arbitrary dimension are permitted.

Słowa kluczowe: discrete multivalued dynamical system, cubical spaces, multivalued maps without continuous selector, Conley index, commutativity property
References

[1] Batko B.; Weak index pairs and the Conley index for discrete multivalued dynamical systems. Part II: properties of the Index, SIAM Journal on Applied Dynamical Systems, 16(3), 2017, pp. 1587–1617.

[2] Batko B., Mischaikow K., Mrozek M., Przybylski M.; Conley index approach to sampled dynamics, in review.

[3] Batko B., Mrozek M.; Weak index pairs and the Conley index for discrete multivalued dynamical systems, SIAM Journal on Applied Dynamical Systems, 15(2), 2016, pp. 1143–1162.

[4] G´orniewicz L.; Topological Fixed Point Theory of Multivalued Mappings, 2nd ed., Topological Fixed Point Theory and Its Applications, Vol. 4, Springer Verlag, The Netherlands, 2006.

[5] Kaczynski T., Mischaikow K., Mrozek M.; Computational Homology, Applied Mathematical Sciences, Vol. 157, Springer-Verlag, 2004.

[6] Lacher R. C.; Cell-like mappings and their generalizations., Bulletin of the American Mathematical Society, 83(4), 1977, pp. 495–552.

[7] Engelking R.; General Topology, Sigma series in pure mathematics, Vol. 6, Heldermann Verlag Berlin, 1989.

[8] Mischaikow K., Mrozek M., Pilarczyk P.; Graph approach to the computation of the homology of continuous maps, Foundations of Computational Mathematics, 5(2), 2005, pp. 199–229.

[9] Mrozek M.; Leray functor and cohomological index for discrete dynamical systems, Transactions of the American Mathematical Society, 318(1), 1990, pp. 149–178.

[10] Pilarczyk P.; CHomP software, 2005. Available via http://chomp.rutgers.edu/Projects/Computational_Homology/OriginalCHomP/software/.

[11] Sakai K.; Cell-Like Maps and Related Topics, in: Geometric Aspects of General Topology, Springer Monographs in Mathematics, Springer Tokyo, 2013.

[12] Szymczak A.; A combinatorial procedure for finding isolating neighborhoods and index pairs, Proceedings of the Royal Society of Edinburgh, A127(5), 1997, pp. 1075–1088.