# 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 *C*^{1} if dim *M* ≥ 3 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.

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

**Keywords:**Milnor attractor, Lyapunov stability, generic dynamics.

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