The University of Arizona

Index Theories in Topological Dynamics

Index Theories in Topological Dynamics

Series: Geometry Seminar
Location: Math 402
Presenter: Juergen Aldinger, University of Munich

Given a compact metric space X and a discrete dynamical system
  phi : Z x X -> X  we consider sets M containing X which consist of uniformly almost periodic orbits. In the case where all such orbits are of the same finite type and M*phi is an equicontinuous system we give a numerical and a vector index. Both are generalizations of the genus in the sense of Krasnosel'ski\u{\i} for sets of periodic points. In the case of M not being equicontinuous we show that an index is, in general, not possible. This suggests to weaken the index axioms and we show how this can be done to get some "weaker" indexes in that case. In addition to that we characterize the orbit structure of the sets M in a very general setting and in the case of M*phi being equicontinuous. We close with a list of conjectures that have grown out of our considerations.

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