Geodesic
On existence of an endomorphism of $2$-torus with strictly invariant repeller
Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 19 (2017) no. 1, pp. 60-66.

Voir la notice de l'article provenant de la source Math-Net.Ru

In the article we construct endomorphism $f$ of 2-torus. This endomorphism satisfies an axiom $A$ and has non-wondering set that contains one-dimensional contracting repeller satisfying following properties: 1) $f(\Lambda)= \Lambda$, $f^{-1}(\Lambda)= \Lambda$; 2) $\Lambda$ is locally homeomorphic to the product of the Cantor set and the interval; 3) $T^2\setminus\Lambda$ consist of a countable family of disjoint open disks. The key idea of construction consists in applying the surgery introduced by S. Smale [1] to an algebraic Anosov endomorphism of the two-torus. We present the results of computational experiment that demonstrate correctness of our construction. Suggested construction reveals significant difference between one-dimensional basic sets of endomorphismsand one-dimensional basic sets of corresponding diffoemorphisms. In particular, the result contrasts with the fact that wondering set of axiom $A$-satisfying diffeomorphism consists of a finite number of open disks in case of spaciously situated basic set [2].
Mots-clés : endomorphism, axiom $A$
Keywords: basic set, repeller. 
@article{SVMO_2017_19_1_a5,
     author = {E. D. Kurenkov},
     title = {On existence of an endomorphism of $2$-torus with strictly invariant repeller},
     journal = {\v{Z}urnal Srednevol\v{z}skogo matemati\v{c}eskogo ob\^{s}estva},
     pages = {60--66},
     publisher = {mathdoc},
     volume = {19},
     number = {1},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SVMO_2017_19_1_a5/}
}
TY  - JOUR
AU  - E. D. Kurenkov
TI  - On existence of an endomorphism of $2$-torus with strictly invariant repeller
JO  - Žurnal Srednevolžskogo matematičeskogo obŝestva
PY  - 2017
SP  - 60
EP  - 66
VL  - 19
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SVMO_2017_19_1_a5/
LA  - ru
ID  - SVMO_2017_19_1_a5
ER  - 
%0 Journal Article
%A E. D. Kurenkov
%T On existence of an endomorphism of $2$-torus with strictly invariant repeller
%J Žurnal Srednevolžskogo matematičeskogo obŝestva
%D 2017
%P 60-66
%V 19
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SVMO_2017_19_1_a5/
%G ru
%F SVMO_2017_19_1_a5
E. D. Kurenkov. On existence of an endomorphism of $2$-torus with strictly invariant repeller. Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 19 (2017) no. 1, pp. 60-66. http://geodesic.mathdoc.fr/item/SVMO_2017_19_1_a5/

[1] S. Smale, “Differentiable dynamical systems”, Bull. Amer. Math. Soc., 73:6 (1967), 747-817 | DOI | MR | Zbl

[2] V. Z. Grines, “On topological conjugacy of diffeomorphisms of 2-manifold on one-dimensional orientable basic sets I”, Trudi Moskovskogo matematicheskogo obshestva, 32 (1975), 35–60 (In Russ.) | Zbl

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