{
  "id": "2301.02301",
  "version": "v1",
  "published": "2023-01-05T21:27:55.000Z",
  "updated": "2023-01-05T21:27:55.000Z",
  "title": "Linear response due to singularities",
  "authors": [
    "Stefano Galatolo",
    "Wael Bahsoun"
  ],
  "categories": [
    "math.DS",
    "nlin.CD"
  ],
  "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 $T_\\eps$ 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-01-05T21:27:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A05",
      "37E05"
    ],
    "keywords": [
      "linear response",
      "turning point",
      "singularities",
      "cusp maps",
      "corresponding invariant density"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}