Common zeroes of families of smooth vector fields on surfaces

Morris W. Hirsch

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arXiv · 1506.02185

Common zeroes of families of smooth vector fields on surfaces

Morris W. Hirsch

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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.

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Morris W. Hirsch. 2015-06-06. Common zeroes of families of smooth vector fields on surfaces. https://arxiv.org/abs/1506.02185

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