{
  "id": "1906.04097",
  "version": "v1",
  "published": "2019-06-10T16:16:56.000Z",
  "updated": "2019-06-10T16:16:56.000Z",
  "title": "Fixed points of post-critically algebraic endomorphisms in higher dimension",
  "authors": [
    "Van Tu Le"
  ],
  "comment": "27 pages, 4 figures",
  "categories": [
    "math.DS",
    "math.CV"
  ],
  "abstract": "A holomorphic endomorphism of $\\mathbb{CP}^n$ is post-critically algebraic if its critical hypersurfaces are periodic or preperiodic. This notion generalizes the notion of post-critically finite rational maps in dimension one. We will study the eigenvalues of the differential of such a map at its fixed points. When $n=1$, a well-known fact is that the eigenvalue at a fixed point is either superattracting or repelling. We prove that when $n=2$ the eigenvalues are still either superattracting or repelling. This is an improvement of a result by Mattias Jonsson. When $n\\geq 2$ and the fixed point is outside the post-critical set, we prove that the eigenvalues are repelling. This result improves one which was already obtained by Fornaess and Sibony under a hyperbolicity assumption on the complement of the post-critical set.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-06-10T16:16:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37F99"
    ],
    "keywords": [
      "fixed point",
      "post-critically algebraic endomorphisms",
      "higher dimension",
      "eigenvalue",
      "post-critically finite rational maps"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}