Linear Response for Bernoulli Convolutions

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 Hölder 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 $\lambda \mathrm{.}$