{
  "id": "math/0407330",
  "version": "v5",
  "published": "2004-07-19T19:43:33.000Z",
  "updated": "2006-03-09T19:55:54.000Z",
  "title": "Martingales, endomorphisms, and covariant systems of operators in Hilbert space",
  "authors": [
    "Dorin Ervin Dutkay",
    "Palle E. T. Jorgensen"
  ],
  "comment": "44 pages, LaTeX2e (\"jotart\" document class); v2: A few opening paragraphs were added to the paper; an addition where a bit of the history is explained, and where some more relevant papers are cited. Corrected a typographical error in Proposition 8.1. v3: A few minor additions: More motivation and explanations in the Intro; Remark 3.3 is new; and eleven relevant references/citations are added; v4: corrected and updated bibliography; v5: more bibliography updates and change of LaTeX document class",
  "journal": "J. Operator Theory 58 (2007), no. 2, 269--310.",
  "categories": [
    "math.CA",
    "math.OA"
  ],
  "abstract": "We show that a class of dynamical systems induces an associated operator system in Hilbert space. The dynamical systems are defined from a fixed finite-to-one mapping in a compact metric space, and the induced operators form a covariant system in a Hilbert space of L^2-martingales. Our martingale construction depends on a prescribed set of transition probabilities, given by a non-negative function. Our main theorem describes the induced martingale systems completely. The applications of our theorem include wavelets, the dynamics defined by iterations of rational functions, and sub-shifts in symbolic dynamics. In the theory of wavelets, in the study of subshifts, in the analysis of Julia sets of rational maps of a complex variable, and, more generally, in the study of dynamical systems, we are faced with the problem of building a unitary operator from a mapping r in a compact metric space X. The space X may be a torus, or the state space of subshift dynamical systems, or a Julia set. While our motivation derives from some wavelet problems, we have in mind other applications as well; and the issues involving covariant operator systems may be of independent interest.",
  "revisions": [
    {
      "version": "v5",
      "updated": "2006-03-09T19:55:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42A16",
      "43A65",
      "42A65",
      "47C15",
      "33C50",
      "42C15",
      "42C40",
      "46E22",
      "47B32",
      "60J15",
      "60G42"
    ],
    "keywords": [
      "hilbert space",
      "covariant system",
      "compact metric space",
      "julia set",
      "endomorphisms"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 44,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......7330E"
    }
  }
}