Linear response for skew-product maps with contracting fibres

TL;DR

This paper develops a new approach to linear response in skew-product systems with contracting fibres, enabling analysis of invariant measures and their parameter sensitivity, with applications to Bernoulli convolutions and solenoidal attractors.

Contribution

It introduces a sectional transfer operator framework for skew-products, establishing existence, uniqueness, and differentiability of invariant measures under broad conditions.

Findings

Linear response established for Bernoulli convolutions.

Invariant measures are differentiable with respect to parameters.

Applicable to hyperbolic systems with intermittency.

Abstract

We study linear response for families of skew-product dynamical systems with contracting fibres. Our approach is based on a sectional transfer operator acting on families of probability measures along the fibres. The operator allows to describe invariant measures of the skew-product in terms of sample measures over the base dynamics, regardless of invertibility or non-invertibility of the base map. Under general assumptions, we establish existence and uniqueness of invariant sample measures and prove their differentiability, with respect to system parameters, in suitable topologies. As an application we obtain linear response for Bernoulli convolutions, which are of prime importance in the study of number theoretic problems and fractals. Another application of our results yields linear response for physical measures of solenoidal attractors with intermittency, an example of a hyperbolic system which cannot be handled by traditional transfer operator techniques.