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Topological Wiener–Wintner ergodic theorems via non-abelian Lie group extensions

Published online by Cambridge University Press:  01 October 2007

SARA I. SANTOS  and
CHARLES WALKDEN
SARA I. SANTOS
Affiliation:
School of Mathematics, The University of Manchester, Oxford Road, Manchester M13 9PL, UK (email: [email protected])
CHARLES WALKDEN
Affiliation:
School of Mathematics, The University of Manchester, Oxford Road, Manchester M13 9PL, UK (email: [email protected])
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Abstract

We generalize a series of topological Wiener–Wintner ergodic theorems due to Walters to the context of group extensions of measure-preserving transformations where the group is a non-abelian compact Lie group. Applications to random ergodic theorems for a shift map are given.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 27 , Issue 5 , October 2007 , pp. 1633 - 1650
Copyright
Copyright © Cambridge University Press 2007

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References

[1]Assani, I.. Wiener–Wintner Ergodic Theorems. World Scientific Publishing, River Edge, NJ, 2003.Google Scholar
[2]Brin, M. I.. The topology of group extensions of C-systems. Math. Notes 18 (1975), 453–465.CrossRefGoogle Scholar
[3]Field, M.. Generating sets for compact semisimple Lie groups. Proc. Amer. Math. Soc. 127 (1999), 3361–3365.CrossRefGoogle Scholar
[4]Furstenberg, H.. Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math. Systems Theory 1 (1967), 1–49.Google Scholar
[5]Keynes, H. B. and Newton, D.. The structure of ergodic measures for compact group extensions. Israel J. Math. 18 (1974), 363–389.Google Scholar
[6]Keynes, H. B. and Newton, D.. Ergodic measures for non-abelian compact group extensions. Israel J. Math. 18 (1974), 363–389.CrossRefGoogle Scholar
[7]Oxtoby, J. C.. Ergodic sets. Bull. Amer. Math. Soc. 58 (1952), 116–136.CrossRefGoogle Scholar
[8]Parry, W.. A cocycle equation for shifts. Symbolic Dynamics and its Applications (Contemporary Mathematics, 135). Ed. P. Walters. American Mathematical Society, Providence, RI, 1992, pp. 327–333.CrossRefGoogle Scholar
[9]Parry, W. and Pollicott, M.. Zeta functions and the periodic orbit structure of hyperbolic dynamics. Astérisque 187–188 (1990).Google Scholar
[10]Parry, W. and Pollicott, M.. The Livsic cocycle equation for compact Lie group extensions of hyperbolic systems. J. London Math. Soc. 56 (1997), 405–416.CrossRefGoogle Scholar
[11]Phelps, R.. Lectures on Choquet’s Theorem. Van Nostrand, New York, 1966.Google Scholar
[12]Robinson, E. A.. On uniform convergence in the Wiener–Wintner theorem. J. London Math. Soc. 49 (1994), 493–501.CrossRefGoogle Scholar
[13]Walters, P.. Topological Wiener–Wintner ergodic theorems and a random L 2 ergodic theorem. Ergod. Th. & Dynam. Sys. 16 (1996), 179–206.CrossRefGoogle Scholar
[14]Wiener, N. and Wintner, A.. Harmonic analysis and ergodic theory. Amer. J. Math. 63 (1941), 415–426.CrossRefGoogle Scholar