The two-dimensional density of Bernoulli Convolutions

TL;DR

This paper explores the complex two-dimensional density structure of Bernoulli convolutions, highlighting combinatorial intricacies and the significance of finite orbits in associated multivalued maps, with references to key prior results and new observations.

Contribution

It introduces the study of the two-dimensional density of Bernoulli convolutions, emphasizing combinatorial structures and finite orbits, and presents new properties and examples.

Findings

Visualization of the density reveals intricate combinatorial patterns.

Finite orbits of multivalued maps play a crucial role in the structure.

New properties and examples of Bernoulli convolutions are discussed.

Abstract

Bernoulli convolutions form a one-parameter family of self-similar measures on the unit interval. We suggest to study their two-dimensional density which has an intricate combinatorial structure. Visualizing this structure we discuss results of Erd\"os, J\'oo, Komornik, Sidorov, de Vries, Jordan, Shmerkin and Solomyak, Feng and Wang. We emphasize the r\^ole of finite orbits of associated multivalued maps and mention a few new properties and examples.