{
  "id": "2010.01803",
  "version": "v1",
  "published": "2020-10-05T06:33:52.000Z",
  "updated": "2020-10-05T06:33:52.000Z",
  "title": "Pro-nilfactors of the space of arithmetic progressions in topological dynamical systems",
  "authors": [
    "Zhengxing Lian",
    "Jiahao Qiu"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "For a topological dynamical system $(X, T)$, $l\\in\\mathbb{N}$ and $x\\in X$, let $N_l(X)$ and $L_x^l(X)$ be the orbit closures of the diagonal point $(x,x,\\ldots,x)$ ($l $ times) under the actions $\\mathcal{G}_{l}$ and $\\tau_l $ respectively, where $\\mathcal{G}_{l}$ is generated by $T\\times T\\times \\ldots \\times T$ ($l $ times) and $\\tau_l=T\\times T^2\\times \\ldots \\times T^l$. In this paper, we show that for a minimal system $(X,T)$ and $l\\in \\mathbb{N}$, the maximal $d$-step pro-nilfactor of $(N_l(X),\\mathcal{G}_{l})$ is $(N_l(X_d),\\mathcal{G}_{l})$, where $\\pi_d:X\\to X/\\mathbf{RP}^{[d]}=X_d,d\\in \\mathbb{N}$ is the factor map and $\\mathbf{RP}^{[d]}$ is the regionally proximal relation of order $d$. Meanwhile, when $(X,T)$ is a minimal nilsystem, we also calculate the pro-nilfactors of $(L_x^l(X),\\tau_l)$ for almost every $x$ w.r.t. the Haar measure. In particular, there exists a minimal $2$-step nilsystem $(Y,T)$ and a countable set $\\Omega\\subset Y$ such that for $y\\in Y\\backslash \\Omega$ the maximal equicontinuous factor of $(L_y^2(Y),\\tau_2)$ is not $(L_{\\pi_1(y)}^2(Y_{1}),\\tau_2)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-05T06:33:52.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "topological dynamical system",
      "arithmetic progressions",
      "step nilsystem",
      "haar measure",
      "diagonal point"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}