Geodesic
On periodic points of torus endomorphisms
Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 21 (2019) no. 4, pp. 480-487.

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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 $C^1$ 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.
Mots-clés : Anosov endomorphism, algebraic toral endomorphism
Keywords: periodic points, semiconjugacy. 
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E. D. Kurenkov; D. I. Mints. On periodic points of torus endomorphisms. Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 21 (2019) no. 4, pp. 480-487. http://geodesic.mathdoc.fr/item/SVMO_2019_21_4_a6/

[1] J. Franks, “Anosov diffeomorphisms on tori”, Proc. Symp. in Pure Math., 14, 1970, 61–93 | DOI | MR | Zbl

[2] S. Newhouse, “On codimension one Anosov diffeomorphisms”, Am. J. Math., 92:3 (1970), 761–770 | DOI | MR | Zbl

[3] A. Manning, “There are no new Anosov diffeomorphisms on tori”, American Journal of Mathematics, 96:3 (1974), 422–429 | DOI | MR | Zbl

[4] F. Przytycki, “Anosov endomorphisms”, Stud. Math., 58:3 (1976), 249–285 | DOI | MR | Zbl

[5] R. Mane, C. Pugh, “Stability of endomorphisms”, Lecture Notes in Math., 468, 1975, 175–184 | DOI | MR | Zbl

[6] M. Zhang, “On the topologically conjugate classes of Anosov endomorphisms on tori”, Chin Ann Math Ser B, 10 (1989), 416–425 | MR | Zbl

[7] A. Katok, B. Hasselblatt, Introduction to the modern theory of dynamical systems, Cambridge University Press, 1995, 824 pp. | MR | Zbl

[8] E. B. Vinberg, Course of algebra, MCNMO, Moscow, 2011, 592 pp. (In Russ)