{
  "id": "1605.04198",
  "version": "v1",
  "published": "2016-05-13T14:49:09.000Z",
  "updated": "2016-05-13T14:49:09.000Z",
  "title": "Degree, mixing, and absolutely continuous spectrum of cocycles with values in compact Lie groups",
  "authors": [
    "Rafael Tiedra de Aldecoa"
  ],
  "comment": "38 pages",
  "categories": [
    "math.DS",
    "math-ph",
    "math.MP",
    "math.SP"
  ],
  "abstract": "We consider skew products $$T_\\phi:X\\times G\\to X\\times G,~~(x,g)\\mapsto(F_1(x),g\\;\\!\\phi(x)),$$ where $X$ is a compact manifold with probability measure, $G$ a compact Lie group with Lie algebra $\\frak g$, $F_1:X\\to X$ the time-one map of a measure-preserving flow, and $\\phi\\in C^1(X,G)$ a cocycle. Then, we define the degree of $\\phi$ as a suitable function $P_\\phi M_\\phi:X\\to\\frak g$, we show that it transforms in a natural way under Lie group homomorphisms and under the relation of $C^1$-cohomology, and we explain how it generalises previous definitions of degree of a cocycle. For each finite-dimensional irreducible representation $\\pi$ of $G$, and $\\frak g_\\pi$ the Lie algebra of $\\pi(G)$, we define in an analogous way the degree of $\\pi\\circ\\phi$ as a suitable function $P_{\\pi\\circ\\phi}M_{\\pi\\circ\\phi}:X\\to\\frak g_\\pi$. If $F_1$ is uniquely ergodic and the functions $\\pi\\circ\\phi$ diagonal, or if $T_\\phi$ is uniquely ergodic, then the degree of $\\phi$ reduces to a constant in $\\frak g$ given by an integral over $X$. As a by-product, we obtain that there is no uniquely ergodic skew product $T_\\phi$ with nonzero degree if $G$ is a connected semisimple compact Lie group. Next, we show that $T_\\phi$ is mixing in the orthocomplement of the kernel of $P_{\\pi\\circ\\phi}M_{\\pi\\circ\\phi}$, and under some additional assumptions we show that $U_\\phi$ has purely absolutely continuous spectrum in that orthocomplement if $(iP_{\\pi\\circ\\phi}M_{\\pi\\circ\\phi})^2$ is strictly positive. Summing up these results for each $\\pi$, one obtains a global result for the mixing and the absolutely continuous spectrum of $T_\\phi$. As an application, we present four explicit cases: when $G$ is a torus, $G=SU(2)$, $G=SO(3,\\mathbb R)$, and $G=U(2)$. In each case, the results we obtain are new, or generalise previous results. Our proofs rely on new results on positive commutator methods for unitary operators.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-05-13T14:49:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A25",
      "37A30",
      "37C40",
      "47A35",
      "58J51"
    ],
    "keywords": [
      "absolutely continuous spectrum",
      "uniquely ergodic",
      "skew product",
      "connected semisimple compact lie group",
      "lie algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}