It is proved that near a compact, invariant, proper subset of a C⁰ flow on a locally compact, connected metric space, at least one, out of twenty eight relevant dynamical phenomena, will necessarily occur. Theorem 1 shows that the connectedness of the phase space implies the existence of a considerably deeper classification of topological flow behaviour in the vicinity of compact invariant sets than that described in the classical theorems of Ura-Kimura and Bhatia. The proposed classification brings...

Let $ℱ$ be a holomorphic one-dimensional foliation on ${\mathbb{P}}^n$ such that the components of its singular locus $\Sigma$ are curves $C_i$ and points $p_j$. We determine the number of $p_j$, counted with multiplicities, in terms of invariants of $ℱ$ and $C_i$, assuming that $ℱ$ is special along the $C_i$. Allowing just one nonzero dimensional component on $\Sigma$, we also prove results on when the foliation happens to be determined by its singular locus.