The spectrum of a perturbation of a hyperbolic toral automorphism

In this paper, we consider a Markov operator (i.e., a contraction preserving the subspace of constants and the nonnegativity of functions) in the $L^2$ space on the $n$-dimensional torus that is a special perturbation of the unitary operator corresponding to a hyperbolic toral automorphism. We prove some properties of its spectrum and the spectrum of some related operators.