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Mixing properties of (α,β)-expansions

Published online by Cambridge University Press:  01 August 2009

KARMA DAJANI ,
YUSUF HARTONO  and
COR KRAAIKAMP
KARMA DAJANI
Affiliation:
Department of Mathematics, Utrecht University, Postbus 80.000, 3508 TA Utrecht, The Netherlands (email: k.dajani1@uu.nl)
YUSUF HARTONO
Affiliation:
Fakultas Keguruan dan Ilmu Pendidikan, Universitas Sriwijaya, Jalan Raya Palembang Prabumulih Km 32, Indralaya 30662, Indonesia (email: y.hartono@unsri.ac.id)
COR KRAAIKAMP
Affiliation:
Delft University of Technology and Thomas Stieltjes Institute for Mathematics, EWI (DIAM), Mekelweg 4, 2628 CD Delft, The Netherlands (email: c.kraaikamp@tudelft.nl)
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Abstract

Let 0<α<1 and β>1. We show that every x∈[0,1] has an expansion of the form where hi=hi(x)∈{0,α/β}, and pi=pi(x)∈{0,1}. We study the dynamical system underlying this expansion and give the density of the invariant measure that is equivalent to the Lebesgue measure. We prove that the system is weakly Bernoulli, and we give a version of the natural extension. For special values of α, we give the relationship of this expansion with the greedy β-expansion.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 4 , August 2009 , pp. 1119 - 1140
Copyright
Copyright © Cambridge University Press 2009

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