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On the construction of smooth ergodic skew-products†

Published online by Cambridge University Press:  19 September 2008

Mahesh G. Nerurkar
Mahesh G. Nerurkar
Affiliation:
Department of Mathematics, The Ohio State University, Columbus, Ohio 43210, USA
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Abstract

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In this paper we prove results about lifting dynamical and ergodic properties of a given smooth dynamical system to its skew-product extensions by smooth cocycles. The classical small divisor argument shows that in general such results are not possible. However, using the notion of the ‘fast periodic approximation’ introduced by A. Katok, we will show that if the dynamical system admits such a ‘fast periodic approximation’ then indeed a certain qualitative behaviour which is prohibited by small divisor type conditions is now in fact generic. The techniques are also applied to show that ‘recurrent-proximal’ behaviour of solutions of linear differential equations with almost periodic coefficients is generic under suitable conditions on the coefficient matrix.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 8 , Issue 2 , June 1988 , pp. 311 - 326
Copyright
Copyright © Cambridge University Press 1988

References

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