There exist transitive piecewise smooth vector fields on $\mathbb{S}^2$ but not robustly transitive

It is well known that smooth (or continuous) vector fields cannot be topologically transitive on the sphere $§^2$. Piecewise-smooth vector fields, on the other hand, may present non-trivial recurrence even on $§^2$. Accordingly, in this paper the existence of topologically transitive piecewise-smooth vector fields on $§^2$ is proved, see Theorem \ref{teorema-principal}. We also prove that transitivity occurs alongside the presence of some particular portions of the phase portrait known as {\it sliding region} and {\it escaping region}. More precisely, Theorem \ref{main:transitivity} 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 $§^2$, see Theorem \ref{main:no-transitive}. We finish the paper proving Theorem \ref{main:general} addressing non-robustness on general compact two-dimensional manifolds.

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