Linear Response for Bernoulli Convolutions

TL;DR

This paper investigates the regularity of a function derived from Bernoulli convolutions, revealing phase transitions in differentiability depending on the parameter lambda.

Contribution

It provides conditions for smoothness and non-smoothness of the function related to Bernoulli convolutions and describes a phase transition in differentiability.

Findings

Function is almost nowhere differentiable for small lambda.

Function is almost everywhere differentiable for large lambda.

Describes phase transition in regularity based on lambda.

Abstract

Let $\mu_{\lambda}$ be the Bernoulli convolution measure with parameter $\lambda\in(0,1)$. We study the regularity of the function %We prove that $h=h_{\phi}:\lambda\mapsto \int_{\mathbb{R}}\phi(x)\,d\mu_{\lambda}(x)$ for H\"older observables $\phi$. We describe sufficient conditions for both smoothness and non smoothness of this function. In particular, we show that for almost every function with respect to certain Wiener like measures on $C[0,1]$, $h_\phi$ exhibits a phase transition: it is almost nowhere differentiable for small $\lambda$ and it is almost everywhere differentiable for large $\lambda.$