{
  "id": "2301.01634",
  "version": "v1",
  "published": "2023-01-04T14:16:12.000Z",
  "updated": "2023-01-04T14:16:12.000Z",
  "title": "Joint spectrum in amenability and self-similarity",
  "authors": [
    "Rongwei Yang"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "The first half of this mostly expository note reviews some notions of joint spectrum of linear operators, and it gives a new characterization of amenable groups in terms of projective spectrum. The second half revisits an application of projective spectrum to the study of self-similar group representations made in [16]. In the case $\\pi$ is the Koopman representation of the infinite dihedral group $D_\\infty$ on the binary tree, it shows that the projective spectrum of $D_\\infty$ coincides with the Julia set of a rational map $F_\\pi: \\mathbb{P}^2\\to \\mathbb{P}^2$ derived from the self-similarity of $\\pi$. This improves the main result in [16].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-01-04T14:16:12.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "joint spectrum",
      "self-similarity",
      "projective spectrum",
      "amenability",
      "self-similar group representations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}