Dominated splitting of differentiable dynamics with C¹-topological weak-star property

Abstract

We study weak hyperbolicity of a differentiable dynamical system which is robustly free of non-hyperbolic periodic orbits of Markus type. Let S be a C¹-class vector field on a closed manifold Mⁿ, which is free of any singularities. It is of C¹-weak-star in case there exists a C¹-neighborhood $¥mathscr{U}$ of S such that for any X ∈ $¥mathscr{U}$, if P is a common periodic orbit of X and S with S_{↾ P} = X_{↾ P}, then P is hyperbolic with respect to X. We show, in the framework of Liao theory, that S possesses the C¹-weak-star property if and only if it has a natural and nonuniformly hyperbolic dominated splitting on the set of periodic points Per(S), for the case n = 3.