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Nearly approximate transitivity (AT) for circulant matrices

  • David Handelman,
    Mathematics Department, University of Ottawa, Ottawa, ON K1N 6N5
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Abstract

By previous work of Giordano and the author, ergodic actions of $\mathbf Z$ (and other discrete groups) are completely classified measure-theoretically by their dimension space, a construction analogous to the dimension group used in C*-algebras and topological dynamics. Here we investigate how far from AT (approximately transitive) can actions be which derive from circulant (and related) matrices. It turns out not very: although non-AT actions can arise from this method of construction, under very modest additional conditions, ATness arises; in addition, if we drop the positivity requirement in the isomorphism of dimension spaces, then all these ergodic actions satisfy an analogue of AT. Many examples are provided.
Keywords: approximately transitive, ergodic transformation, circulant matrix, hemicirculant matrix, dimension space, matrix-valued random walk approximately transitive, ergodic transformation, circulant matrix, hemicirculant matrix, dimension space, matrix-valued random walk
MSC Classifications: 37A05, 06F25, 28D05, 46B40, 60G50 show english descriptions Measure-preserving transformations
Ordered rings, algebras, modules {For ordered fields, see 12J15; see also 13J25, 16W80}
Measure-preserving transformations
Ordered normed spaces [See also 46A40, 46B42]
Sums of independent random variables; random walks
37A05 - Measure-preserving transformations
06F25 - Ordered rings, algebras, modules {For ordered fields, see 12J15; see also 13J25, 16W80}
28D05 - Measure-preserving transformations
46B40 - Ordered normed spaces [See also 46A40, 46B42]
60G50 - Sums of independent random variables; random walks
 

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