Counting β-expansions and the absolute continuity of Bernoulli convolutions

We study the typical growth rate of the number of words of length n which can be extended to β-expansions of x. In the general case we give a lower bound for the growth rate, while in the case that the Bernoulli convolution associated to parameter β is absolutely continuous we are able to give the growth rate precisely. This gives new necessary and sufficient conditions for the absolute continuity of Bernoulli convolutions.