{
  "id": "2109.07941",
  "version": "v1",
  "published": "2021-09-16T12:49:17.000Z",
  "updated": "2021-09-16T12:49:17.000Z",
  "title": "Joint ergodicity of Hardy field sequences",
  "authors": [
    "Konstantinos Tsinas"
  ],
  "comment": "54 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "We study mean convergence of multiple ergodic averages, where the iterates arise from smooth functions of polynomial growth that belong to a Hardy field. Our results include all logarithmico-exponential functions of polynomial growth, such as the functions $t^{3/2}, t\\log t$ and $e^{\\sqrt{\\log t}}$. We show that if all non-trivial linear combinations of the functions $a_1,...,a_k$ stay logarithmically away from rational polynomials, then the $L^2$-limit of the ergodic averages $\\frac{1}{N} \\sum_{n=1}^{N}f_1(T^{\\lfloor{a_1(n)}\\rfloor}x)\\cdots f_k(T^{\\lfloor{a_k(n)}\\rfloor}x)$ exists and is equal to the product of the integrals of the functions $f_1,...,f_k$ in ergodic systems, which establishes a conjecture of Frantzikinakis. Under some more general conditions on the functions $a_1,...,a_k$, we also find characteristic factors for convergence of the above averages and deduce a convergence result for weak-mixing systems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-09-16T12:49:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A45",
      "28D05",
      "05D10",
      "11B30"
    ],
    "keywords": [
      "hardy field sequences",
      "joint ergodicity",
      "polynomial growth",
      "study mean convergence",
      "non-trivial linear combinations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 54,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}