Preprint A114/2001
Attractors of generic diffeomorphims are persistent

Flavio Abdenur

**Keywords: **
Generic properties | homoclinic classes | transitive attractors

We prove that, $C^1$-generically (that is, for every diffeomorphism in a $C^1$-residual subset of $\text{Diff}^1(M)$), every $\Omega$-isolated transitive set (in particular, every transitive attracting set) is persistent in a $C^1$-locally residual sense. This implies that such sets admit weakly hyperbolic dominated splittings in arbitrary dimensions, and in particular that they are hyperbolic in the case of surface diffeomorphisms.
We also prove that, generically, an $\Omega$-isolated transitive set is either hyperbolic or else approached by diffeomorphisms exhibiting a heterodimensional cycle, a type of homoclinic bifurcation.