### Conley index over the base morphism for multivalued discrete dynamical systems

DOI: http://dx.doi.org/10.12775/TMNA.2007.007

#### Abstract

We define an index of Conley type for a certain class of

uppersemicontinuous multivalued dynamical systems, using

techniques introduced by Mrozek, Reineck and Srzednicki [< i> The Conley index over a base< /i> , Trans.

Amer. Math. Soc. < b> 352< /b> (2000), no. 9, 4171–4194]

for the index over the base. We give the characterisation of the

nontrivial index and present an example, proving that our index

detects isolated invariant sets that are not detected by

Kaczyński and Mrozek's [< i> Conley index for discrete multi-valued dynamical systems< /i> ,

Topology Appl. < b> 65< /b> (1995), 83–96] index.

uppersemicontinuous multivalued dynamical systems, using

techniques introduced by Mrozek, Reineck and Srzednicki [< i> The Conley index over a base< /i> , Trans.

Amer. Math. Soc. < b> 352< /b> (2000), no. 9, 4171–4194]

for the index over the base. We give the characterisation of the

nontrivial index and present an example, proving that our index

detects isolated invariant sets that are not detected by

Kaczyński and Mrozek's [< i> Conley index for discrete multi-valued dynamical systems< /i> ,

Topology Appl. < b> 65< /b> (1995), 83–96] index.

#### Keywords

Conley index; isolated invariant set; index pair; multivalued dynamical system

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