There Exist Transitive Piecewise Smooth Vector Fields on $\mathbb {S}^2$ but Not Robustly Transitive

Abstract

It is well known that smooth (or continuous) vector fields cannot be topologically transitive on the sphere $\mathbb {S}^2$. Piecewise-smooth vector fields, on the other hand, may present nontrivial recurrence even on $\mathbb {S}^2$. Accordingly, in this paper the existence of topologically transitive piecewise-smooth vector fields on $\mathbb {S}^2$ is proved (see Theorem A). We also prove that transitivity occurs alongside the presence of some particular portions of the phase portrait known as sliding region and esca** region. More precisely, Theorem B states that, under the presence of transitivity, trajectories must interchange between sliding and esca** regions through tangency points. In addition, we prove that every transitive piecewise-smooth vector field is neither robustly transitive nor structural stable on $\mathbb {S}^2$ (see Theorem C). We finish the paper proving Theorem D addressing non-robustness on general compact two-dimensional manifolds.