Common zeroes of families of smooth vector fields on surfaces

Abstract

Let Y and X denote $$C^k$$ vector fields on a possibly noncompact surface with empty boundary, $$1\le k <\infty $$ . Say that Y tracks X if the dynamical system it generates locally permutes integral curves of X. Let K be a locally maximal compact set of zeroes of X. Theorem Assume the Poincare–Hopf index of X at K is nonzero, and the k-jet of X at each point of K is nontrivial. If $$\mathfrak {g}$$ is a supersolvable Lie algebra of $$C^k$$ vector fields that track X, then the elements of $$\mathfrak {g}$$ have a common zero in K. Applications are made to the dynamics of attractors and transformation groups.