题目：On the Changing of the Dimension of a Basic Set by Means of Simple Bifurcations

报告人：Olga Pochinka 教授（下诺夫哥罗德市高等经济学校）

地点：宁静楼110室

时间：11月1日（星期三），下午 3:30-4:30

报告摘要

One of the fifty problems of dynamical systems formulated by J. Palis and C. Pugh is the problem of the existence of a simple arc joining two structural stable systems on a smooth closed manifold [1]. Simplicity means that the arc consists of structural stable systems with the exception of a finite set of points where the system has the least deviation from a rough system, that is simple bifurcations: a saddle-node, a doubling period or a bifurcation of a heteroclinic tangency. We show that by means of such arc it is possible to change the dimension of a basic set. Namely we joint arbitrary structural stable 3-diffeomorphism with an expanding two-dimensional attractor with an Anosov diffeomorphism.

In dimension 3, the problem of the existence of a simple arc is complicated by the presence of diffeomorphisms with wildly embedded separatrices. The first ``wild'' example was constructed by D. Pixton [2]. This diffeomorphism belongs to the class consisting of three-dimensional Morse-Smale diffeomorphisms for which the non-wandering set consists of exactly four points: two sinks, a source and a saddle. As follows from the paper by Ch. Bonatti, V. Grines, V. Medvedev, O. Pochinka [3], two Pixton diffeomorphisms can be joined by a simple arc. This effect is due to the fact that for any diffeomorphism from the Pixton class at least one one-dimensional separatrix of the saddle point is tame [4].

Due to the paper by V. Grines, E. Zhuzhoma [5] that only the 3-torus admits a rough 3-diffeomorphism f with an expanding two-dimensional attractor and all the basic set of f are trivial except the attractor. We check that all separatrices of the trivial basic sets and the boundary points are tame. It allows us to put all these separatrices to a smooth arc and confluence saddles and nodes.

参考文献：[1] Palis J., Pugh C.,Fifty problems in dynamical systems. 1974 [2] Pixton D.,Wild unstable manifolds. 1977 [3] Bonatti Ch., Grines V., Medvedev V., Pochinka O.,Bifurcations of diffeomorphisms Morse-Smale with wildly embedded separatrices. 2007 [4] Bonatti Ch., Grines V., Knots as topological invariant for gradient-like diffeomorphisms ofthe 3-sphere. 2000 [5] Grines V., Zhuzhoma E.,On structurally stable diffeomorphisms with codimension one expanding attractors 2005

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