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January  2007, 18(1): 199-217. doi: 10.3934/dcds.2007.18.199

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, Netherlands, Netherlands

Received  June 2006 Revised  November 2006 Published  February 2007

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: Bernoulli shifts., greedy and lazy expansions.
Mathematics Subject Classification: Primary, 37A05, 11K5.
Citation: Karma Dajani, Charlene Kalle. Random β-expansions with deleted digits. Discrete & Continuous Dynamical Systems, 2007, 18 (1) : 199-217. doi: 10.3934/dcds.2007.18.199
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