Common zeroes of families of smooth vector fields on surfaces

TL;DR

This paper investigates the conditions under which a family of smooth vector fields on surfaces share common zeroes, especially focusing on the role of the Poincaré-Hopf index and tracking properties.

Contribution

It establishes a theorem linking the Poincaré-Hopf index and tracking vector fields to the existence of common zeroes on surfaces.

Findings

Nonzero Poincaré-Hopf index implies common zero for tracking vector fields.

Conditions on the k-jet ensure the triviality of local behavior at zeroes.

Applications include attractors and transformation groups on surfaces.

Abstract

Let Y and X denote C^k vector fields on a possibly noncompact surface with empty boundary, k >0. 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 Poincar'e-Hopf index of X at K is nonzero, and the k-jet of X at each point of K is nontrivial. If g is a supersolvable Lie algebra of C^k vector fields that track X, then the elements of g have a common zero in K. Applications are made to attractors and transformation groups.