Note on spectra of non-selfadjoint operators over dynamical systems
  We consider equivariant continuous families of discrete one-dimensional operators over arbitrary dynamical systems. We introduce the concept of a pseudo-ergodic element of a dynamical system. We then show that all operators associated to pseudo-ergodic elements have the same spectrum and that this spectrum agrees with their essential spectrum. As a consequence we obtain that the spectrum is constant and agrees with the essential spectrum for all elements in the dynamical system if minimality holds. 
Siegfried Beckus, Daniel Lenz, Marko Lindner, Christian Seifert
math.SP math-ph math.FA math.MP