ISSN 2079-6900 (Print) 
ISSN 2587-7496 (Online)

Middle Volga Mathematical Society Journal

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MSC2010 37C70

On existence of an endomorphism of $2$-torus with strictly invariant repeller

E. D. Kurenkov1

AnnotationIn the article we construct an axiom $A$ endomorphism $f$ of 2-torus with nonwondering 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 \cite{Sm} to an algebraic 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 of endomorphisms and one-dimensional basic sets of corresponding diffoemorphisms. In particular, the result contrasts with the fact that wondering set of axiom $A$ diffeomorphism consist of a finite number of open disks in case of spaciously situated basic set \cite{Gr75}.
Keywordsendomorphism, axiom $A$, basic set, repeller

1Evgeniy D. Kurenkov, research assistant of laboratory TMD, National Research University Higher School of Economics (25 Bolshaya Pechyorskaya Str., Nizhnii Novgorod 603155, Russia), ORCID:,

Citation: E. D. Kurenkov, "[On existence of an endomorphism of $2$-torus with strictly invariant repeller]", Zhurnal Srednevolzhskogo matematicheskogo obshchestva,19:1 (2017) 60–66 (In Russian)

DOI 10.15507/2079-6900.19.2017.01.60-66