Locally topologically generic diffeomorphisms with Lyapunov unstable Milnor attractors

TL;DR

This paper demonstrates that in certain smooth dynamical systems, Lyapunov unstable Milnor attractors are topologically generic, especially near homoclinic tangencies, with implications for stability in various dimensions and regularity classes.

Contribution

It establishes the topological genericity of Lyapunov unstable Milnor attractors in smooth manifolds near homoclinic tangencies, extending understanding of attractor instability in dynamical systems.

Findings

Unstable Milnor attractors are topologically generic near homoclinic tangencies.

Instability of Milnor attractors is generic in $C^1$ for dimension ≥ 3 and in $C^2$ for dimension 2.

Either homoclinic classes admit dominated splittings or have unstable Milnor attractors in generic diffeomorphisms.

Abstract

We prove that for every smooth compact manifold $M$ and any $r \ge 1$, whenever there is an open domain in $\mathrm{Diff}^r(M)$ 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 $\mathrm{dim}\,M \ge 3$ and in $C^2$ if $\mathrm{dim}\,M = 2$. Moreover, it follows from the results of C. Bonatti, L. J. D\'iaz 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.