Heterodimensional tangencies

Enrique R. Pujals | Diaz, Lorenzo | Nogueira, Adriana

Keywords:

We consider $C^1$-diffeomorphisms $f$ defined on a closed three dimensional manifold with a pair of saddles $P_f$ and $Q_f$ (of indices one and two) whose homoclinic classes coincide. We prove that if the two dimensional stable manifold of $P_f$ and the two dimensional unstable manifold of $Q_f$ have some non-transverse intersection (a heterodimensional tangency) the unfolding of such a tangency leads to diffeomorphism $h$ such that the homoclinic class of $Q_h$ is robustly non-dominated. This leads to the phenomena of ($C^1$-locally generic) coexistence of infinitely many sinks or sources and, in some cases, also to the coexistence of infinitely many minimal Cantor sets. We also give examples where the previous dynamical configuration occurs and provides a natural transition from partially hyperbolic to robustly non-dominated dynamics.

Anexos:

• dnp-final.ps