Bernoulli Convolutions and Salem Numbers (Morlet Chair Shigeki Akiyama)

Bernoulli convolutions are probably the simplest and most well studied examples of self-similar measures with overlaps. They form a family of self-similar measures, each associated to some parameter lambda, but despite extensive study over the years there are very few classes of lambda for which we know whether the corresponding Bernoulli convolution is absolutely continuous.

We shall study Bernoulli convolutions associated to Salem numbers by first constructing a countable Markov shift associated with the corresponding set of finite polynomials, and then studying the structure of this shift and its recurrence properties. This project lies at the interface between dynamical systems, fractal geometry and number theory.