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Common zeroes of families of smooth vector fields on surfaces (1506.02185v1)

Published 6 Jun 2015 in math.DS

Abstract: Let Y and X denote Ck 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 Ck vector fields that track X, then the elements of g have a common zero in K. Applications are made to attractors and transformation groups.

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