Jose Ferreira Alves

A smooth path through a rough landscape

We address the linear response problem for Bernoulli
convolutions across the entire parameter range, including both
absolutely continuous and singular invariant measures. Our approach
relies on a geometric representation via a fat baker map, viewed as a
skew-product system with contracting fibres. We introduce a sectional
transfer operator that acts on sections of probability measures on the
fibres and show that it is a contraction in the Wasserstein-1
distance. This yields the existence and uniqueness of a fixed point
corresponding to the physical invariant distribution. We further prove
a smooth dependence of this fixed point on the parameter in a suitable
Banach space, establishing the linear response uniformly across the
regular and singular regimes. Our approach also provides a linear
response for solenoidal attractors with intermittency. Joint work with
Wael Bahsoun.