Linear response for systems with a cusp

TL;DR

This paper demonstrates that linear response theory applies to a class of tent-like dynamical systems with cusps, extending previous results to systems with singularities and providing formulas in $L^p$ spaces.

Contribution

It extends linear response results to systems with cusp singularities, broadening the class of maps where linear response is valid.

Findings

Linear response holds for tent-like maps with cusps.

Results extend previous work to systems with singularities.

Linear response formula derived in $L^p$ spaces for $p>1$.

Abstract

In this note we consider a tent-like family with a cusp at the singular point and show that the linear response holds for certain perturbations of this family. This contrasts the tent-like maps with finite derivatives at the singularity. Our results extend the results of Bahsoun and Galatolo to the larger class of singularities and we obtain the linear response formula in $L^p$ for $p>1$.