Linear response due to singularities

. It is well known that a family of tent-like maps with bounded derivatives has no linear response for typical deterministic perturbations chang- ing the value of the turning point. In this note we prove a rather unexpected result: if we consider a tent-like family with a cusp at the turning point, we re- cover the linear response. More precisely, let T ε 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 ε be the corresponding invariant density. We prove that ε (cid:55)→ h ε is differentiable in L 1 and provide a formula for its derivative.

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