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arXiv · 2301.02301

Linear response due to singularities

Wael Bahsoun·Stefano Galatolo
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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.

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Wael Bahsoun, Stefano Galatolo. 2023-01-05. Linear response due to singularities. https://arxiv.org/abs/2301.02301

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