Random β-expansions with deleted digits

Karma Dajani; Charlene Kalle

doi:10.3934/dcds.2007.18.199

Article Contents

2007, Volume 18, Issue 1: 199-217. Doi: 10.3934/dcds.2007.18.199

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Random β-expansions with deleted digits

Karma Dajani^1, and
Charlene Kalle^1,

1.
Department of Mathematics, Utrecht University, Postbus 80.000, 3508 TA Utrecht

Received: May 31, 2006

Revised: October 31, 2006

Published: December 2006

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Abstract

In this paper we define random $\beta$-expansions with digits taken from a given set of real numbers $A= \{ a_1 , \ldots , a_m \}$. We study a generalization of the greedy and lazy expansion and define a function $K$ that generates essentially all $\beta$-expansions with digits belonging to the set $A$. We show that $K$ admits an invariant measure $\nu$ under which $K$ is isomorphic to the uniform Bernoulli shift on $A$.

Keywords:

greedy and lazy expansions,
Bernoulli shifts.

Mathematics Subject Classification: Primary, 37A05, 11K55.

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