Bernoulli Convolutions and 1D Dynamics

TL;DR

This paper introduces a family of dynamical systems that preserve Bernoulli convolutions and explores conditions under which these systems are piecewise convex, linking convexity to the absolute continuity of the convolutions.

Contribution

It proposes a new family of dynamical systems on the interval that preserve Bernoulli convolutions and investigates their convexity properties related to measure regularity.

Findings

Piecewise convexity of the systems implies absolute continuity with bounded density.

Numerical evidence suggests specific parameter ranges for convexity.

The systems provide a new perspective on the structure of Bernoulli convolutions.

Abstract

We describe a family $\phi_{\lambda}$ of dynamical systems on the unit interval which preserve Bernoulli convolutions. We show that if there are parameter ranges for which these systems are piecewise convex, then the corresponding Bernoulli convolution will be absolutely continuous with bounded density. We study the systems $\phi_{\lambda}$ and give some numerical evidence to suggest values of $\lambda$ for which $\phi_{\lambda}$ may be piecewise convex.