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

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 Schr\"odinger 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 Schr\"odinger operators and the fibered rotation number; and the Gap Labelling Theorem via the Schwartzman group, as expected from the one dimensional case [AS1983, JM1982].

Categories: math.DS math-ph math.MP math.SP