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Mosc. Math. J., 2017, Volume 17, Number 3, Pages 511–553 (Mi mmj645)  

Locally topologically generic diffeomorphisms with Lyapunov unstable Milnor attractors

Ivan Shilin

Moscow Center for Continuous Mathematical Education, Bolshoy Vlasyevskiy per., 11, Moscow, Russia, 119002

Abstract: We prove that for every smooth compact manifold $M$ and any $r\ge1$, whenever there is an open domain in $\operatorname{Diff}^r(M)$ exhibiting a persistent homoclinic tangency related to a basic set with a sectionally dissipative periodic saddle, topologically generic diffeomorphisms in this domain have Lyapunov unstable Milnor attractors. This implies, in particular, that the instability of Milnor attractors is locally topologically generic in $C^1$ if $\dim M\ge3$ and in $C^2$ if $\dim M=2$. Moreover, it follows from the results of C. Bonatti, L. J. Díaz and E. R. Pujals that, for a $C^1$ topologically generic diffeomorphism of a closed manifold, either any homoclinic class admits some dominated splitting, or this diffeomorphism has an unstable Milnor attractor, or the inverse diffeomorphism has an unstable Milnor attractor. The same results hold for statistical and minimal attractors.

Key words and phrases: Milnor attractor, Lyapunov stability, generic dynamics.

Funding Agency Grant Number
Simons Foundation
Russian Foundation for Basic Research 16-01-00748-a
This research was supported in part by the Simons Foundation and by the RFBR grant 16-01-00748-a.


Full text: http://www.mathjournals.org/.../2017-017-003-005.html
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Document Type: Article
MSC: Primary 37B25; Secondary 37B20, 37C20, 37C29, 37D30
Received: May 24, 2016; in revised form June 2, 2017
Language: English

Citation: Ivan Shilin, “Locally topologically generic diffeomorphisms with Lyapunov unstable Milnor attractors”, Mosc. Math. J., 17:3 (2017), 511–553

Citation in format AMSBIB
\Bibitem{Shi17}
\by Ivan~Shilin
\paper Locally topologically generic diffeomorphisms with Lyapunov unstable Milnor attractors
\jour Mosc. Math.~J.
\yr 2017
\vol 17
\issue 3
\pages 511--553
\mathnet{http://mi.mathnet.ru/mmj645}


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