In this thesis, we consider CMV matrices, Jacobi matrices and Schr"{o}dinger operators while assuming that the coefficients and potentials are generated by dynamical systems.

One of the major parts investigates continuous cocycles arising from CMV and Jacobi matrices. Assuming the Verblunsky and Jacobi coefficients arise from generalized skew-shifts, we prove that uniform hyperbolicity of the associated cocycles is C0-dense. This implies that the associated CMV and Jacobi matrices have Cantor spectrum for a generic continuous sampling map.

Another major part concerns the Fibonacci Hamiltonian. In the classical Fibonacci Hamiltonian, a sampling function is a locally constant function of a very special form. In this thesis, we study whether spectral results of the classical Fibonacci Hamiltonian can be extended to more general sampling functions. We provide the trace map description of the spectrum and extend the results for the classical Fibonacci Hamiltonian to arbitrary locally constant sampling functions.