Minimally Nonhyperbolic Sets for Diffeomorphisms C1 away from Homoclinic Bifurcations

Abstract

Palis conjectures that diffeomorphisms with a homoclinic tangency or a heterodimensional cycle are C¹ dense in the complement of the C¹ closure of hyperbolic diffeomorphisms. We prove some results towards the conjecture: C¹ away from homoclinic tangencies and heterodimensional cycles, a C¹ generic diffeomorphism must have no simple minimally nonhyperbolic sets, and any nonsimple minimally nonhyperbolic set, if exists, must be very special.