Homoclinic tangencies and dominated splittings

Abstract

We characterize diffeomorphisms that are C1 away from homoclinic tangencies by using dominated splittings on the C1 preperiodic sets. More precisely, we prove that a diffeomorphism f cannot be C1 approximated by systems with homoclinic tangencies if and only if for all 1⩽i⩽d−1, the C1 i-preperiodic set P*i(f) of f has i-dominated splitting.