Moscow Mathematical Journal

Volume 17, Issue 3, July–September 2017 pp. 511–553.

Locally Topologically Generic Diffeomorphisms with Lyapunov Unstable Milnor Attractors

Authors: Ivan Shilin (1)
Author institution:(1) Moscow Center for Continuous Mathematical Education, Bolshoy Vlasyevskiy per., 11, Moscow, Russia, 119002

Summary:

We prove that for every smooth compact manifold M and any r ≥ 1, whenever there is an open domain in 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 C¹ if dim M ≥ 3 and in C² if dim M = 2. Moreover, it follows from the results of C. Bonatti, L. J. Díaz and E.R. Pujals that, for a C¹ 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.

2010 Math. Subj. Class. Primary: 37B25; Secondary: 37B20, 37C20, 37C29, 37D30.

Keywords: Milnor attractor, Lyapunov stability, generic dynamics.

Contents Full-Text PDF