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Dynamics in Siberia - 2019
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On one-dimensional basic sets of endomorphisms of 2-torus E. D. Kurenkov |
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Abstract: In 1967, Smale [1] proposed a method for constructing nontrivial basis sets based on Anosov diffeomorphism. In fact, Smale schematically described a surgery operation on the Anosov diffeomorphism, which results in a DA-diffeomorphism with basis sets having a topological dimension one less than the dimension of an ambient manifold. The idea of surgery is the following. In the small neighborhood of a fixed saddle point the saddle fixed point is replaced by three fixed points: two saddles and one node. In this case, the following two scenarios are equally possible: 1) the node is a source; 2) the node is a sink. In the first case, the nonwandering set of the DA-diffeomorphism consists of a source fixed point and one-dimensional expanding attractor. In the second case the nonwandering set consists of a sink fixed point and one-dimensional contracting repeller. In [2] a class of not one-to-one smooth maps generalizing Anosov diffeomorphisms was introduced. It seems quite natural to consider surgery operation for algebraic endomorphisms, for example, for Anosov endomorphism \begin{equation} \left\{\begin{aligned}\overline x&=3x+y\\\overline y&=x+y\end{aligned}\right.\quad\text{mod }1. \end{equation} Since In [3], the second scenario of Smale’s surgery operation for algebraic Anosov endomorphism of type (1) was numerically implemented. As a result, the The main result of this report is that the first scenario for Anosov endomorphism (1) does not lead to an Theorem. Let
Acknowledgments. Research is done with financial support of Russian Science Foundation (project 17-11-01041). Bibliography [1] Smale S. Differentiable dynamical systems //Bulletin of the American mathematical Society. 1967, Vol. 73., No. 6., pp. 747–817. [2] Przytycki F. Anosov endomorphisms //Studia mathematica. 1976, Vol. 3., no. 58., pp. 249–285. [3] Kurenkov E. D. On existence of of endomorphism of 2-torus with strictly invariant contracting repeller // Zhurnal Srednevolzhskogo Matematicheskogo Obshchestva. 2017, Vol.19, no. 1. pp. 60–66. [4] Grines V.Z., Zhuzhoma E.V., Kurenkov E.D. Surgery operation for Anosov endomorphisms of 2-torus does not lead to expanding attractor // Dynamical systems. 2018, Vol. 8(36), no. 3, pp. 235–244. Language: English |