MSC2010 37D05
On periodic points of torus endomorphisms
E. D. Kurenkov1, D. I. Mints2
| Annotation | It is a well-known fact that Anosov endomorphisms of n-torus which are different from automorphisms and expanding endomorphisms are not structurally stable and, in general, are not conjugated to algebraic endomorphisms. Nevertheless, hyperbolic algebraic endomorphisms of torus are conjugated with their C1 perturbations on the set of periodic points. Therefore the study of algebraic toral endomorphisms is very important. This paper is devoted to study of the structure of the sets of periodic and pre-periodic points of algebraic toral endomorphisms. Various group properties of this sets of points are studied. The density of periodic points for algebraic endomorphisms of n-torus is proved; it is clarifief how the number of periodic and pre-periodic points with a fixed denominator depends on the properties of the characteristic polynomial. The Theorem 1.1 is the main result of this paper. It contains an algorithm that allows to split the sets of periodic and pre-periodic points of a given algebraic endomorphism of two-dimensional torus. |
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| Keywords | Anosov endomorphism, algebraic toral endomorphism, periodic points, semiconjugacy |
1Evgeniy D. Kurenkov, Doctoral Student, Department of Fundamental Mathematics, Research Assistant of Laboratory of Topological Methods in Dynamics, National Research University Higher School of Economics in Nizhny Novgorod (25/12, Bolshaya Pecherskaya Str., Nizhny Novgorod, 603155, Russia), ORCID: https://orcid.org/0000-0002-3544-1143, ekurenkov@hse.ru
2Dmitriy I. Mints, Research Assistant of Laboratory of Topological Methods in Dynamics, National Research University Higher School of Economics in Nizhny Novgorod (25/12, Bolshaya Pecherskaya Str., Nizhny Novgorod, 603155, Russia), ORCID: https://orcid.org/0000-0003-0329-6946, dmincz@hse.ru
Citation: E. D. Kurenkov, D. I. Mints, "[On periodic points of torus endomorphisms]", Zhurnal Srednevolzhskogo matematicheskogo obshchestva,21:4 (2019) 480–487 (In Russian)
DOI 10.15507/2079-6900.21.201904.480-487
