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| Format: | Preprint |
| Veröffentlicht: |
2026
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2604.20883 |
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| _version_ | 1866915950559756288 |
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| author | Fu, Jianning |
| author_facet | Fu, Jianning |
| contents | Let $μ_λ$ be the Bernoulli convolution measure with parameter $λ\in(0,1)$. We study the regularity of
the function
%We prove that
$h=h_ϕ:λ\mapsto \int_{\mathbb{R}}ϕ(x)\,dμ_λ(x)$
for Hölder observables $ϕ$. 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 $C[0,1]$,
$h_ϕ$ exhibits a phase transition: it is almost nowhere differentiable for small
$λ$ and it is almost everywhere differentiable for large $λ.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_20883 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Linear Response for Bernoulli Convolutions Fu, Jianning Dynamical Systems Classical Analysis and ODEs Let $μ_λ$ be the Bernoulli convolution measure with parameter $λ\in(0,1)$. We study the regularity of the function %We prove that $h=h_ϕ:λ\mapsto \int_{\mathbb{R}}ϕ(x)\,dμ_λ(x)$ for Hölder observables $ϕ$. 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 $C[0,1]$, $h_ϕ$ exhibits a phase transition: it is almost nowhere differentiable for small $λ$ and it is almost everywhere differentiable for large $λ.$ |
| title | Linear Response for Bernoulli Convolutions |
| topic | Dynamical Systems Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2604.20883 |