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  • 1 Utrecht University Department of Mathematics Postbus 80.000 3508 TA Utrecht the Netherlands
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We construct a planar version of the natural extension of the piecewise linear transformation T generating greedy β-expansions with digits in an arbitrary set of real numbers A = {a0, a1, a2}. As a result, we derive in an easy way a closed formula for the density of the unique T-invariant measure µ absolutely continuous with respect to Lebesgue measure. Furthermore, we show that T is exact and weak Bernoulli with respect to µ.

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