Linear response due to singularities

arXiv2301.02301MaRDI QIDQ6508356

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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 a rather unexpected result: if we consider a tent-like family with a emph{cusp} at the turning point, we recover the linear response. More precisely, let $T_{e}ps$ 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 $h_{e}ps$ be the corresponding invariant density. We prove that $epsmapsto{h}_{e}ps$ is differentiable in $L^1$ and provide a formula for its derivative.

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