{
  "id": "1705.00159",
  "version": "v1",
  "published": "2017-04-29T09:21:54.000Z",
  "updated": "2017-04-29T09:21:54.000Z",
  "title": "Boundedness and absoluteness of some dynamical invariants in model theory",
  "authors": [
    "Krzysztof Krupinski",
    "Ludomir Newelski",
    "Pierre Simon"
  ],
  "categories": [
    "math.LO",
    "math.DS",
    "math.GN"
  ],
  "abstract": "Let ${\\mathfrak C}$ be a monster model of an arbitrary theory $T$, $\\bar \\alpha$ any tuple of bounded length of elements of ${\\mathfrak C}$, and $\\bar c$ an enumeration of all elements of ${\\mathfrak C}$. By $S_{\\bar \\alpha}({\\mathfrak C})$ denote the compact space of all complete types over ${\\mathfrak C}$ extending $tp(\\bar \\alpha/\\emptyset)$, and $S_{\\bar c}({\\mathfrak C})$ is defined analogously. Then $S_{\\bar \\alpha}({\\mathfrak C})$ and $S_{\\bar c}({\\mathfrak C})$ are naturally $Aut({\\mathfrak C})$-flows. We show that the Ellis groups of both these flows are of bounded size (i.e. smaller than the degree of saturation of ${\\mathfrak C}$), providing an explicit bound on this size. Next, we prove that these Ellis groups do not depend on the choice of the monster model ${\\mathfrak C}$; thus, we say that they are absolute. We also study minimal left ideals (equivalently subflows) of the Ellis semigroups of the flows $S_{\\bar \\alpha}({\\mathfrak C})$ and $S_{\\bar c}({\\mathfrak C})$. We give an example of a NIP theory in which the minimal left ideals are of unbounded size. We show that in each of these two cases, boundedness of a minimal left ideal is an absolute property (i.e. it does not depend on the choice of ${\\mathfrak C}$) and that whenever such an ideal is bounded, then its isomorphism type is also absolute. Assuming NIP, we give characterizations of when a minimal left ideal of the Ellis semigroup of $S_{\\bar c}({\\mathfrak C})$ is bounded. Then we adapt a proof of Chernikov and Simon to show that whenever such an ideal is bounded, the natural epimorphism (described by Krupinski, Pillay and Rzepecki) from the Ellis group of the flow $S_{\\bar c}({\\mathfrak C})$ to the Kim-Pillay Galois group $Gal_{KP}(T)$ is an isomorphism (in particular, $T$ is G-compact). We provide some counter-examples for $S_{\\bar \\alpha}({\\mathfrak C})$ in place of $S_{\\bar c}({\\mathfrak C})$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-04-29T09:21:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03C45",
      "54H20",
      "37B05"
    ],
    "keywords": [
      "model theory",
      "dynamical invariants",
      "ellis group",
      "boundedness",
      "ellis semigroup"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}