{
  "id": "1611.05257",
  "version": "v1",
  "published": "2016-11-16T12:37:32.000Z",
  "updated": "2016-11-16T12:37:32.000Z",
  "title": "Mating quadratic maps with the modular group II",
  "authors": [
    "Shaun Bullett",
    "Luna Lomonaco"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "In 1994 S. Bullett and C. Penrose introduced the one complex parameter family of $(2:2)$ holomorphic correspondences $\\mathcal{F}_a$: $$\\left(\\frac{aw-1}{w-1}\\right)^2+\\left(\\frac{aw-1}{w-1}\\right)\\left(\\frac{az+1}{z+1}\\right) +\\left(\\frac{az+1}{z+1}\\right)^2=3$$ and proved that for every value of $a \\in [4,7] \\subset \\mathbb{R}$ the correspondence $\\mathcal{F}_a$ is a mating between a quadratic polynomial $Q_c(z)=z^2+c,\\,\\,c \\in \\mathbb{R}$ and the modular group $\\Gamma=PSL(2,\\mathbb{Z})$. They conjectured that this is the case for every member of the family $\\mathcal{F}_a$ which has $a$ in the connectedness locus. We prove here that every member of the family $\\mathcal{F}_a$ which has $a$ in the connectedness locus is a mating between the modular group and an element of the parabolic quadratic family $Per_1(1)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-11-16T12:37:32.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "modular group",
      "mating quadratic maps",
      "connectedness locus",
      "parabolic quadratic",
      "quadratic polynomial"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}