Video library: V. Z. Grines, On the topology of manifolds admitting cascades attractor-repeller of the same dimension

VIDEO LIBRARY


Dynamics in Siberia - 2019 February 28, 2019 12:00, 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
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). Bibliography [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