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arXiv · 2604.20883

Linear Response for Bernoulli Convolutions

Jianning Fu
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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 $\lambda.$

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Jianning Fu. 2026-04-16. Linear Response for Bernoulli Convolutions. https://arxiv.org/abs/2604.20883

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