Growth rate for beta-expansions.
Втр, 08/18/2009 - 11:53 — SomNambul
link: http://arxiv.org/abs/0902.0488
Abstract
Abstract
Let $\beta>1$ and let $m>\be$ be an integer. Each $x\in
I_\be:=[0,\frac{m-1}{\beta-1}]$ can be represented in the form \[
x=\sum_{k=1}^\infty \epsilon_k\beta^{-k}, \] where
$\epsilon_k\in\{0,1,...,m-1\}$ for all $k$ (a $\beta$-expansion of $x$). It is
known that a.e. $x\in I_\beta$ has a continuum of distinct $\beta$-expansions.
In this paper we prove that if $\beta$ is a Pisot number, then for a.e. $x$
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
$\beta$.
When $\beta<\frac{1+\sqrt5}2$, we show that the set of $\beta$-expansions
grows exponentially for every internal $x$.
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