Linear response due to singularities

TL;DR

This paper demonstrates that tent-like maps with a cusp at the turning point exhibit linear response to deterministic perturbations, unlike traditional tent maps with bounded derivatives.

Contribution

It proves differentiability of the invariant density with respect to perturbations for cusp tent maps and provides an explicit formula for the derivative.

Findings

Invariant density is differentiable in L^1 for cusp tent maps.

Explicit formula for the derivative of the invariant density.

Linear response is recovered in cusp maps, unlike in bounded derivative tent maps.

Abstract

It is well known that a family of tent-like maps with bounded derivatives has no linear response for typical deterministic perturbations changing the value of the turning point. In this note we prove the following result: if we consider a tent-like family with a \emph{cusp} at the turning point, we recover the linear response. More precisely, let $T_\eps$ be a family of such cusp maps generated by changing the value of the turning point of $T_0$ by a deterministic perturbation and let $h_\eps$ be the corresponding invariant density. We prove that $\eps\mapsto h_\eps$ is differentiable in $L^1$ and provide a formula for its derivative.