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 and a cocycle on this flow, , then is called *close to linear* if 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 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.

**1.**Hillel Furstenberg, Harvey B. Keynes, Nelson G. Markley, and Michael Sears,*Topological properties of 𝑅ⁿ suspensions and growth properties of 𝑍ⁿ cocycles*, Proc. London Math. Soc. (3)**66**(1993), no. 2, 431–448. MR**1199074**, 10.1112/plms/s3-66.2.431**2.**H. B. Keynes, and M. Sears,*Time changes for flows and suspensions*, Pacific J Math., 130 No 1 (1987), 97-113.**3.**H. B. Keynes, N. G. Markley, and M. Sears,*On the structure of minimal 𝑅ⁿ actions*, Quaestiones Math.**16**(1993), no. 1, 81–102. MR**1217478****4.**H. B. Keynes, N. G. Markley, and M. Sears,*Ergodic averages and integrals of cocycles*, Acta Math. Univ. Comemanae LXIV (1995), 123--139.

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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

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