On existence and uniqueness of the carrying simplex for competitive dynamical systems

Certain dynamical models of competition have a unique invariant hypersurface to whichevery nonzero tractory is asymptotic, having simple geometry and topology.Comment: Submitted to Journal of Biological Dynamics. 13 page

Common zeros of inward vector fields on surfaces

A vector field X on a manifold M with possibly nonempty boundary is inward if it generates a unique local semiflow $\Phi^X$. A compact relatively open set K in the zero set of X is a block. The Poincar\'e-Hopf index is generalized to an index for blocks that may meet the boundary. A block with nonzero index is essential. Let X, Y be inward $C^1$ vector fields on surface M such that $[X,Y]\wedge X=0$ and let K be an essential block of zeros for X. Among the main results are that Y has a zero in K if X and $Y$ are analytic, or Y is $C^2$ and $\Phi^Y$ preserves area. Applications are made to actions of Lie algebras and groups

Zero sets of Lie algebras of analytic vector fields on real and complex 2-manifolds

Let X be an analytic vector field on a real or complex 2-manifold, and K a compact set of zeros of X whose fixed point index is not zero. Let A denote the Lie algebra of analytic vector fields Y on M such that at every point of M the values of X and [X,Y] are linearly dependent. Then the vector fields in A have a common zero in K. Application: Let G be a connected Lie group having a 1-dimensional normal subgroup. Then every action of G on M has a fixed point.Comment: 22 page