Bernoulli Convolutions and Measures on the Spectra of Algebraic Integers

Abstract: Given an algebraic integer $\beta$ and alphabet $A=\{-1,0,1\}$, the spectrum of $\beta$ is the set $$\Sigma(\beta) :=\bigg\{\sum_{i=1}^n a_i\beta^i : n\in\mathbb N, a_i\in A\bigg\}.$$ In the case that $\beta$ is Pisot one can study the spectrum of $\beta$ dynamically using substitutions or cut and project schemes, and this allows one to see lots of local structure in the spectrum. There are higher dimensional analogues for other algebraic integers.

In this talk we will define a random walk on the spectrum of $\beta$ and show how, with appropriate renormalisation, this leads to an infinite stationary measure on the spectrum. This measure has local structure analagous to that of the spectrum itself. Furthermore, this measure has deep links with the Bernoulli convolution, and in particular new criteria for the absolute continuity of Bernoulli convolutions can be stated in terms of the ergodic properties of these measures.

dynamical systemsnumber theory

Audience: researchers in the topic

One World Numeration seminar

Series comments: Description: Online seminar on numeration systems and related topics

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