Abstract
In [20] the author and A. Vershik have shown that for β=1/2(1 + √5) and the alphabet {0,1} the infinite Bernoulli convolution (= the Erdős measure) has a property similar to the Lebesgue measure. Namely, it is quasi-invariant of type II1 under the β-shift, and the natural extension of the β-shift provided with the measure equivalent to the Erdős measure, is Bernoulli. In this note we extend this result to all Pisot parameters β (modulo some general arithmetic conjecture) and an arbitrary “sufficient” alphabet.
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