There Exist Transitive Piecewise Smooth Vector Fields on but Not Robustly Transitive

It is well known that smooth (or continuous) vector fields cannot be topologically transitive on the sphere S-2 . Piecewise-smooth vector fields, on the other hand, may present nontrivial recurrence even on S-2. Accordingly, in this paper the existence of topologically transitive piecewise-smooth vector fields on 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 escaping region. More precisely, Theorem B states that, under the presence of transitivity, trajectories must interchange between sliding and escaping regions through tangency points. In addition, we prove that every transitive piecewise-smooth vector field is neither robustly transitive nor structural stable on S-2 (see Theorem C). We finish the paper proving Theorem D addressing non-robustness on general compact two-dimensional manifolds. (AU)