Let \( (\Theta ,T,\mu ) \) be an ergodic topological dynamical system. The fibered rotation number for cocycles in \( \Theta \times \textrm{SL}(2,{{\mathbb {R}}}) \) , 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 \( \Theta \times \textrm{Sp}(2\,m,{{\mathbb {R}}}) \) or \( \Theta \times \textrm{HSp}(2\,m,\mathbb {C}) \) , where \( \textrm{Sp}(2m,{{\mathbb {R}}}) \) and \( \textrm{HSp}(2m,\mathbb {C}) \) 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 [11, 33].
