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$\beta$-transformation, natural extension and invariant measure

Published online by Cambridge University Press:  10 November 2000

GAVIN BROWN  and
QINGHE YIN
GAVIN BROWN
Affiliation:
Vice-Chancellor, The University of Sydney, NSW 2006, Australia (e-mail: [email protected])
QINGHE YIN
Affiliation:
School of Mathematics and Statistics, The University of Sydney, NSW 2006, Australia (e-mail: [email protected])
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Abstract

For $\beta>1$, consider the $\beta$-transformation $T_\beta$. When $\beta$ is an integer, the natural extension of $T_\beta$ can be represented explicitly as a map on the unit square with an invariant measure: the corresponding two-dimensional Lebesgue measure. We show that, under certain conditions on $\beta$, the natural extension is defined on a simply connected region and an invariant measure is a constant multiple of the Lebesgue measure.

We characterize those $\beta$ in terms of the $\beta$-expansion of one, and study the structure and size of the set of all such $\beta$.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 20 , Issue 5 , October 2000 , pp. 1271 - 1285
Copyright
© 2000 Cambridge University Press

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