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Dynamics in Siberia - 2019
February 28, 2019 12:00–12:50, Novosibirsk, Sobolev Institute of Mathematics of Russian Academy of Sciences, Conference Hall

Plenary talks

On the topology of manifolds admitting cascades attractor-repeller of the same dimension

V. Z. Grines

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V. Z. Grines
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Abstract: Let $f:M^n\rightarrow M^n$ be orientation preserving diffeomorphism of a smooth closed orientable manifold $M^n$ satisfying axiom $A$ of S.Smale (non-wandering set $\mathrm{NW}(f)$ is hyperbolic and the set of periodic points is dense in $\mathrm{NW}(f)$).
According to S.Smale's spectral theorem, the nonwandering set $\mathrm{NW}(f)$ can be decomposed into a finite union of disjoint closed invariant sets (called basic sets), each of which contains a dense orbit.
It is well known that if the non-wandering set of diffeomorphism $f$ consists of exactly two fixed points: a source and a sink, then the manifold $M^n$ is diffeomorphic to the $n$-dimensional sphere $\mathbb S^n$. If the dimension of a basic set of diffeomorphism $f$ coincides with the dimension of the ambient manifold, then $f$ is Anosov diffeomorphism, the basic set is an attractor and a repeller simultaneously and coincides with the manifold $M^n$. It was shown by J.Franks and Sh.Newhouse in the case when the dimension of a stable or unstable manifold of a periodic point of Anosov diffeomorphism is 1, that manifold $M^n$ is diffeomorphic to the torus of dimension $n$ (see [1, 2]).
The report describes the results obtained in the works of V.S.Grines, E.V.Zhuzhoma, Yu.A.Levchenko, V.Medvedev, O.Pochinka (see [3–5]), from which follows topological classification of manifolds $M^n$ admitting diffeomorphisms $f$ whose nonwandering sets consist of an attractor and a repeller of the same dimension. In addition, we give sufficient conditions when the nonwandering set of the diffeomorphism $f$ cannot consist of two basic sets of the same dimension.
The report is prepared with the financial support of the Russian Science Foundation (project 17-11-01041).
[1] J.Franks, Anosov diffeomorphisms, In: Global Analisys, Proc. Symp. in Pure Math., 14, 61–93 (1970). [2] S.Newhouse, On codimension one Anosov diffeomorphisms, Am. J. Math., 92, No. 3, 761–770 (1970).
[3] V.Grines, Yu.Levchenko, V.S.Medvedev, and O.Pochinka, The topological classification of structural stable 3-diffeomorphisms with two-dimensional basic sets, Nonlinearity, 28, 4081–4102 (2015).
[4] V.Grines, T.Medvedev, O.Pochinka, Dynamical systems on 2-and 3-manifolds. Switzerland. Springer International Publishing, 2016.
[5] V.Z.Grines, Ye.V.Zhuzhoma, O.V.Pochinka. Rough diffeomorphisms with basic sets of codimension one. Journal of Mathematical Sciences, Vol. 225, No. 2, August, 2017.

Language: English

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