Locally topologically generic diffeomorphisms with Lyapunov unstable Milnor attractors

zbMATH Open1416.37028arXiv1604.02437MaRDI QIDQ5383940

Publication date: 20 June 2019

Abstract: We prove that for every smooth compact manifold

M

and any

r g e 1

, whenever there is an open domain in

m a t h r m {D i f f}^{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

m a t h r m d i m, M g e 3

and in

C^{2}

if

m a t h r m d i m, 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.

Full work available at URL: https://arxiv.org/abs/1604.02437

zbMATH Keywords

Lyapunov stability Milnor attractor generic dynamics

Mathematics Subject Classification ID

Attractors and repellers of smooth dynamical systems and their topological structure (37C70) Generic properties, structural stability of dynamical systems (37C20) Partially hyperbolic systems and dominated splittings (37D30) Stability theory for smooth dynamical systems (37C75) Homoclinic and heteroclinic orbits for dynamical systems (37C29) Dynamical systems involving smooth mappings and diffeomorphisms (37C05)