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β-transformation, invariant measure and uniform distribution

Part of: Probabilistic theory: distribution modulo $1$; metric theory of algorithms Measure-theoretic ergodic theory

Published online by Cambridge University Press:  09 April 2009

Gavin Brown  and
Qinghe Yin
Gavin Brown
Affiliation:
Vice-Chancellor, The University of Sydney, NSW 2006, Australia
Qinghe Yin
Affiliation:
School of Mathematics and Statistics, The University of Sydney, NSW 2006, Australia
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Abstract

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Let Tβ be the β-transformation on [0, 1). When β is an integer Tβ is ergodic with respect to Lebesgue measure and almost all orbits {} are uniformly distributed. Here we consider the non-integer case, determine when Tα, Tβ have the same invariant measure and when (appropriately normalised) orbits are uniformly distributed.

Keywords

β-transformationinvariant measureuniform distribution

MSC classification

Secondary: 28D05: Measure-preserving transformations 11K55: Metric theory of other algorithms and expansions; measure and Hausdorff dimension 11K06: General theory of distribution modulo $1$
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 66 , Issue 3 , June 1999 , pp. 418 - 428
Copyright
Copyright © Australian Mathematical Society 1999

References

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