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SKEW PRODUCTS OF SHIFTS WITH A COMPACT LIE GROUP

Published online by Cambridge University Press:  01 October 1997

WILLIAM PARRY
WILLIAM PARRY
Affiliation:
Mathematics Institute, Warwick University, Coventry CV4 7AL
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Abstract

We consider aperiodic shifts of finite type σ with an equilibrium state m and associated skew-products σf where f: X→G is Hölder and G is a compact Lie group. We show that generically σf is weak-mixing and give constructive methods for achieving weak-mixing by perturbing an arbitrary f at a finite number of small neighbourhoods. When σf is not ergodic we describe the (closed) ergodic decomposition precisely. The key result shows that certain measurable eigenfunctions are essentially Hölder continuous. This leads to conditions for weak-mixing or ergodicity in terms of a functional equation which involves Hölder rather than measurable functions.

Type
Notes and Papers
Information
Journal of the London Mathematical Society , Volume 56 , Issue 2 , October 1997 , pp. 395 - 404
Copyright
The London Mathematical Society 1997

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