Asymptotic pairs in topological actions of amenable groups.

We provide a definition of a ≺-asymptotic (we suggest the pronunciation "prec-asymptotic") pair in a topological action (X , G) of a countable amenable group G , where ≺ is an order on G of type Z. In the case where for some G -invariant Borel probability measure μ on X , the measure-preserving system (X , μ , G) factors, via a map φ , onto a multiorder (O ˜ , ν , G) , we also introduce the notion of a φ -asymptotic pair. Then we prove that if μ has positive measure-theoretic conditional entropy with respect to the multiorder factor, then the set of points which belong to φ -asymptotic pairs has positive measure μ. This result is a generalization of the Blanchard-Host-Ruette Theorem for classical topological dynamical systems (actions of Z). As a strengthening of our theorem, we show that for any system (X , G) of positive topological entropy, any multiorder (O ˜ , ν , G) and ν -almost every ≺ ∈ O ˜ , there exist ≺-asymptotic pairs in X. Finally, we characterize systems (X , G) of topological entropy zero as factors of topologically multiordered systems (in which case φ is defined μ -almost everywhere for every G -invariant measure μ) with no φ -asymptotic pairs. [ABSTRACT FROM AUTHOR]