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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On close to linear cocycles


Authors: H. B. Keynes, N. G. Markley and M. Sears
Journal: Proc. Amer. Math. Soc. 124 (1996), 1923-1931
MSC (1991): Primary 58F25; Secondary 28D10, 54H20
MathSciNet review: 1307537
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Abstract: If we have a flow $(X,\Bbb {Z}^m)$ and a cocycle $h$ on this flow, $h:X\times \Bbb {Z}^m\rightarrow \Bbb {R}^m$, then $h$ is called close to linear if $h$ can be written as the direct sum of a linear (constant) cocycle and a cocycle in the closure of the coboundaries. Many of the desirable consequences of linearity hold for such cocycles and, in fact, a close to linear cocycle is cohomologous to a cocycle which is norm close to a linear one. Furthermore in the uniquely ergodic case all cocycles are close to linear. We also establish that a close to linear cocycle which is covering is cohomologous to one with the special property that it can be extended by piecewise linearity to an invertible cocycle from $X\times \Bbb {R}^m$ to itself. This implies that a suspension obtained from a close to linear cocycle is isomorphic to a time change of the suspension obtained from the identity cocycle.


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Additional Information

H. B. Keynes
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
Email: keynes@math.umn.edu

N. G. Markley
Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742
Email: ngm@glve.umd.edu

M. Sears
Affiliation: Department of Mathematics, University of the Witwatersrand, Johannesburg, South Africa
Email: 036mis@cosmos.wits.ac.za

DOI: http://dx.doi.org/10.1090/S0002-9939-96-03188-7
PII: S 0002-9939(96)03188-7
Received by editor(s): February 25, 1994
Received by editor(s) in revised form: November 11, 1994
Communicated by: Linda Keen
Article copyright: © Copyright 1996 American Mathematical Society