Finite orbits in multivalued maps and Bernoulli convolutions

TL;DR

This paper investigates the structure of Bernoulli convolutions, focusing on the role of finite orbits in multivalued maps, their relation to singularities, and how these orbits depend on the parameter, revealing complex network-like structures.

Contribution

It introduces a detailed analysis of finite orbits in multivalued maps associated with Bernoulli convolutions, linking them to measure singularities and parameter-dependent structures.

Findings

Finite orbits outside the overlap region exist for all a0>1.6182.

Orbits intersecting the overlap region form network-like structures at Perron parameters.

Singularities are related to intersections of address curves, including non-Pisot and non-Salem parameters.

Abstract

Bernoulli convolutions are certain measures on the unit interval depending on a parameter $\beta$ between 1 and 2. In spite of their simple definition, they are not yet well understood. We study their two-dimensional density which exists by a theorem of Solomyak. To each Bernoulli convolution, there is an interval $D$ called the overlap region, and a map which assigns two values to each point of $D$ and one value to all other points of $[0,1].$ There are two types of finite orbits of these multivalued maps which correspond to zeros and potential singularities of the density, respectively. Orbits which do not meet $D$ belong to an ordinary map called $\beta$-transformation and exist for all $\beta>1.6182.$ They were studied by Erd\"os, J\'oo, Komornik, Sidorov, de Vries and others as points with unique addresses, and by Jordan, Shmerkin and Solomyak as points with maximal local dimension. In the two-dimensional view, these orbits form address curves related to the Milnor-Thurston itineraries in one-dimensional dynamics. The curves depend smoothly on the parameter and represent quantiles of all corresponding Bernoulli convolutions. Finite orbits which intersect $D$ have a network-like structure and can exist only at Perron parameters $\beta.$ Their points are intersections of extended address curves, and can have finite or countable number of addresses, as found by Sidorov. For an uncountable number of parameters, the central point $\frac12$ has only two addresses. The intersection of periodic address curves can lead to singularities of the measures. We give examples which are not Pisot or Salem parameters. It seems that all singularities of Bernoulli convolutions are related to network-like orbits. The paper is self-contained and includes many illustrations.