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Nearly Approximate Transitivity (AT) for Circulant Matrices

Part of: Stochastic processes Ordered structures Ergodic theory Normed linear spaces and Banach spaces; Banach lattices Measure-theoretic ergodic theory

Published online by Cambridge University Press:  07 March 2019

David Handelman
David Handelman*
Affiliation:
Mathematics Department, University of Ottawa, Ottawa, ON K1N 6N5 Email: dehsg@uottawa.ca
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Abstract

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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 $\text{C}^{\ast }$-algebras and topological dynamics. Here we investigate how far from approximately transitive (AT) actions can be that 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, approximate transitivity arises. KIn 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 transitiveergodic transformationcirculant matrixhemicirculant matrixdimension spacematrix-valued random walk

MSC classification

Primary: 37A05: Measure-preserving transformations
Secondary: 06F25: Ordered rings, algebras, modules 28D05: Measure-preserving transformations 46B40: Ordered normed spaces 60G50: Sums of independent random variables; random walks
Type
Article
Information
Canadian Journal of Mathematics , Volume 71 , Issue 2 , April 2019 , pp. 381 - 415
Copyright
© Canadian Mathematical Society 2019 

Footnotes

Supported in part by an NSERC Discovery Grant.

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