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

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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. 
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     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/}
}
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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/