Quenched and annealed linear response for some partially hyperbolic skew products

TL;DR

This paper establishes statistical stability and linear response for a class of partially hyperbolic skew products with parameter-dependent base maps, extending previous results to new cases.

Contribution

It introduces novel formulas for linear response and derivatives of asymptotic moments when both fiber and base maps depend on parameters.

Findings

Proves quenched and annealed stability for these systems.

Derives explicit formulas for linear response in the new setting.

Extends applicability to partially hyperbolic maps not covered before.

Abstract

We prove quenched and annealed statistical stability, linear response, and differentiability of asymptotic moments for parametric families of partially hyperbolic skew products, with random hyperbolic maps on the fibers. The main novelty is that the base maps also depend on the parameter, which leads to different formulas in the linear response and the derivative of the asymptotic moments with respect to the parameter. Our annealed results apply to partially hyperbolic maps that are not covered in \cite{BashCastro26,Dol,DS}.