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Mixing and asymptotic distribution modulo 1

Published online by Cambridge University Press:  19 September 2008

Russell Lyons
Russell Lyons
Affiliation:
Department of Mathematics, Stanford University, Stanford CA 94305, USA
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Abstract

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If μ is a probability measure which is invariant and ergodic with respect to the transformation x↦qx on the circle ℝ/ℤ, then according to the ergodic theorem, {qnx} has the asymptotic distribution μ for μ-a.e. x. On the other hand, Weyl showed that when μ is Lebesgue measure, λ, and {mj} is an arbitrary sequence of integers increasing strictly to ∞, the asymptotic distribution of {mjx} is λ for λ-a.e. x. Here, we investigate the asymptotic distributions of {mjx} μ-a.e. for fairly arbitrary {mj} under some strong mixing conditions on μ. The result is a kind of stable ergodicity: the distributions are obtained from simple operations applied to μ. The ideas extend to the situation of a sequence of transformations x↦qnx where invariance is not present. This gives us information about many Riesz products and Bernoulli convolutions. Finally, we apply the theory to resolve some questions about H-sets.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 8 , Issue 4 , December 1988 , pp. 597 - 619
Copyright
Copyright © Cambridge University Press 1988

References

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