Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-12-05T02:26:04.570Z Has data issue: false hasContentIssue false
  • English
  • Français

$\beta$-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])
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)