The giant spiral
type: Item
instance of: scholarly article
title: The giant spiral
zbMATH Open document ID: 0917.58039
DOI: 10.1007/BF02762268
published in: Israel Journal of Mathematics
publication date: 1999-08-05T00:00:00Z
review text: Cocycles of \(\mathbb{Z}^m\) actions on compact metric spaces provide a means for constructing \(\mathbb{R}^m\) actions or flows, called suspension flows. It is known that all \(\mathbb{R}^m\) flows with a free dense orbit have an almost one-to-one extension which is a suspension flow. When \(m=1\), examples of cocycles are easy to construct; there is a one-to-one correspondence between cocycles and real valued continuous functions. However, when \(m>1\), the construction of examples of cocycles becomes more problematic. The only existing class of examples, the close to linear cocycles, have strong linearity properties and are well understood. In fact, when the \(\mathbb{Z}^m\) action is uniquely ergodic, all cocycles are close to linear. We show here that in general this need not be the case. We present a method, suggested to us by Hillel Furstenberg, for constructing examples of cocycles when \(m>1\) and use this method to construct a non close to linear cocycle on a minimal \(\mathbb{Z}^2\) action.
Mathematics Subject Classification ID: 37C10
Mathematics Subject Classification ID: 37A99
zbMATH DE Number: 1250640
zbMATH Keywords: actions
zbMATH Keywords: suspension flow
zbMATH Keywords: cocycles
zbMATH Keywords: flows
author: Kathleen Madden
reviewed by: Messoud Efendiev
MaRDI profile type: Publication
cites work: Topological Properties of Rn Suspensions and Growth Properties of Zn Cocycles
cites work: On close to linear cocycles
cites work: Q4261780
cites work: Q4261754
full work available at URL: https://doi.org/10.1007/bf02762268
OpenAlex ID: W2033340413
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description: scientific article; zbMATH DE number 1250640