The fibered rotation number for ergodic symplectic cocycles and its applications: I. Gap Labelling Theorem

Abstract

Let $(\Theta ,T,\mu )$ be an ergodic topological dynamical system. The fibered rotation number for cocycles in $\Theta \times \mathrm{SL}(2,\mathbb{R})$, acting on $\Theta \times \mathbb{R}{\mathbb{P}}^1$ is well-defined and has wide applications in the study of the spectral theory of Schrödinger operators. In this paper, we will provide its natural generalization for higher dimensional cocycles in $\Theta \times \mathrm{SP}(2m,\mathbb{R})$ or $\Theta \times \mathrm{HSP}(2m,\mathbb{C})$, where $\mathrm{SP}(2m,\mathbb{R})$ and $\mathrm{HSP}(2m,\mathbb{C})$ respectively refer to the $2m$-dimensional symplectic or Hermitian-symplectic matrices. As a corollary, we establish the equivalence between the integrated density of states for generalized Schrödinger operators and the fibered rotation number; and the Gap Labelling Theorem via the Schwartzman group, as expected from the one dimensional case [AS1983, JM1982].

25 pages, revised version