{
  "id": "2301.10620",
  "version": "v1",
  "published": "2023-01-25T14:55:45.000Z",
  "updated": "2023-01-25T14:55:45.000Z",
  "title": "Absolute continuity of self-similar measures on the plane",
  "authors": [
    "Boris Solomyak",
    "Adam Śpiewak"
  ],
  "categories": [
    "math.DS",
    "math.CA",
    "math.PR"
  ],
  "abstract": "Consider an iterated function system consisting of similarities on the complex plane of the form $g_{i}(z) = \\lambda_i z + t_i,\\ \\lambda_i, t_i \\in \\mathbb{C},\\ |\\lambda_i|<1, i=1,\\ldots, k$. We prove that for almost every choice of $(\\lambda_1, \\ldots, \\lambda_k)$ in the super-critical region (with fixed translations and probabilities), the corresponding self-similar measure is absolutely continuous. This extends results of Shmerkin-Solomyak (in the homogenous case) and Saglietti-Shmerkin-Solomyak (in the one-dimensional non-homogeneous case). As the main steps of the proof, we obtain results on the dimension and power Fourier decay of random self-similar measures on the plane, which may be of independent interest.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-01-25T14:55:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A78",
      "28A80",
      "37A46",
      "42A38"
    ],
    "keywords": [
      "absolute continuity",
      "power fourier decay",
      "random self-similar measures",
      "complex plane",
      "one-dimensional non-homogeneous case"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}