Uniform ergodic theorems for discontinuous skew-product flows and applications to Schrödinger equations

Motivated by linear Schrödinger equations with almost periodic potentials and phase transitions over almost periodic lattices, we introduce the so-called skew-product quasi-flows (SPQFs), which may admit both temporal and spatial discontinuity. In this paper we establish two basic theorems for SPQFs. One is an extension of the Bogoliubov–Krylov theorem for the existence of invariant Borel probability measures and the other is the uniform ergodic theorems. As applications, it will be shown that such a Schrödinger equation admits a well-defined rotation number