Linear Response for Bernoulli Convolutions

Abstract

Let $\hat{I}{¼}_{\hat{I}}»$ be the Bernoulli convolution measure with parameter $\hat{I}»\in (0,1)$. We study the regularity of the function %We prove that $h=h_{\ddot{I}}•:\hat{I}»\mapsto {\int }_{\mathbb{R}}\ddot{I}•(x)d\hat{I}{¼}_{\hat{I}}»(x)$ for Hölder observables $\ddot{I}•$. 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 $C[0,1]$, $h_{\ddot{I}}•$ exhibits a phase transition: it is almost nowhere differentiable for small $\hat{I}»$ and it is almost everywhere differentiable for large $\hat{I}».$