We show that for C 1 generic diffeomorphisms, an isolated homoclinic class is shadow-able if and only if it is a hyperbolic basic set.
We show that C 1-generically, a differentiable map is positively measure expansive if and only if it is expanding.
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.
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