Weak measure expansivity of C 2 dynamics

Abstract

Abstract Let f f be a C 2 {C}^{2} -diffeomorphism with Axiom A and no cycle condition on a two-dimensional smooth manifold. In this article, we prove that if f f is C 2 {C}^{2} -robustly weak measure expansive, then it is Q 2 {Q}^{2} -Anosov. Moreover, we expand the results of the C 2 {C}^{2} -diffeomorphism case into the C 2 {C}^{2} -vector field on a three-dimensional smooth manifold. Let X X be a C 2 {C}^{2} -vector field with Axiom A and no cycle condition. We prove that if X X is C 2 {C}^{2} -robustly weak measure expansive, then it is Q 2 {Q}^{2} -Anosov.