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Growth rate for beta-expansions
Let and let m>\be be an integer. Each x\in
I_\be:=[0,\frac{m-1}{\beta-1}] can be represented in the form where
for all (a -expansion of ). It is
known that a.e. has a continuum of distinct -expansions.
In this paper we prove that if is a Pisot number, then for a.e.
this continuum has one and the same growth rate. We also link this rate to the
Lebesgue-generic local dimension for the Bernoulli convolution parametrized by
.
When , we show that the set of -expansions
grows exponentially for every internal .Comment: 21 pages, 2 figure
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