Dominated splitting of differentiable dynamics with $\mathrm{C}^1$-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 $\mathrm{C}^1$-class vector field on a closed manifold $M^n$, which is free of any singularities. It is of $\mathrm{C}^1$-weak-star in case there exists a $\mathrm{C}^1$-neighborhood $\mathscr{U}$ of S such that for any X$\in\mathscr{U}$, if $P$ is a common periodic orbit of X and S with S$_{\upharpoonright P}=$X$_{\upharpoonright P}$, then $P$ is hyperbolic with respect to X. We show, in the framework of Liao theory, that S possesses the $\mathrm{C}^1$-weak-star property if and only if it has a natural and nonuniformly hyperbolic dominated splitting on the set of periodic points $\mathrm{Per}$(S), for the case $n=3$.