Geodesic
The spectrum of a~perturbation of a~hyperbolic toral automorphism
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXII, Tome 411 (2013), pp. 125-134

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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. 
@article{ZNSL_2013_411_a7,
     author = {A. M. Levin},
     title = {The spectrum of a~perturbation of a~hyperbolic toral automorphism},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {125--134},
     publisher = {mathdoc},
     volume = {411},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2013_411_a7/}
}
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A. M. Levin. The spectrum of a~perturbation of a~hyperbolic toral automorphism. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXII, Tome 411 (2013), pp. 125-134. http://geodesic.mathdoc.fr/item/ZNSL_2013_411_a7/