Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Topological mapping properties defined by digraphs

Pages: 83 - 92, Volume 5, Issue 1, January 1999      doi:10.3934/dcds.1999.5.83

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John Banks - Department of Mathematics, La Trobe University Bundoora, Australia 3083, Australia (email)

Abstract: Topological transitivity, weak mixing and non-wandering are definitions used in topological dynamics to describe the ways in which open sets feed into each other under iteration. Using finite directed graphs, these definitions are generalized to obtain topological mapping properties. The extent to which these mapping properties are logically distinct is examined. There are three distinct properties which entail "interesting" dynamics. Two of these, transitivity and weak mixing, are already well known. The third does notappear in the literature but turns out to be close to weak mixing in a sense to be discussed. The remaining properties comprise a countably infinite collection of distinct properties entailing somewhat less interesting dynamics and including non-wandering.

Keywords:  Topological transitivity, weak mixing, non-wandering.
Mathematics Subject Classification:  54H20.

Received: February 1998;      Revised: July 1998;      Available Online: October 1998.