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Search Results: 1 - 10 of 86633 matches for " Morris W. Hirsch "
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On existence and uniqueness of the carrying simplex for competitive dynamical systems
Morris W. Hirsch
Mathematics , 2008,
Abstract: Certain dynamical models of competition have a unique invariant hypersurface to whichevery nonzero tractory is asymptotic, having simple geometry and topology.
Zero sets of Lie algebras of analytic vector fields on real and complex 2-manifolds
Morris W. Hirsch
Mathematics , 2013,
Abstract: 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.
Jacobians and branch points of real analytic open maps
Morris W. Hirsch
Mathematics , 2005,
Abstract: The main result is that the Jacobian determinant of an analytic open map from Euclidean n-space to itself does not change sign. A corollary of the proof is that the set of branch points has dimension < n-1.
Periodic orbits and homoclinic loops for surface homeomorphisms
Morris W. Hirsch
Mathematics , 2005,
Abstract: Let p be a saddle fixed point for an orientation-preserving surface diffeomorphism f admitting a homoclinic point q. Let V be an open 2-cell bounded by a simple loop formed by two arcs joining p to q lying respectively in the stable and unstable curves at p. It is shown that f|V has fixed point index 1 or 2 depending only on the geometry of V near p.
Actions of Lie groups and Lie algebras on manifolds
Morris W. Hirsch
Mathematics , 2012,
Abstract: Questions of the following sort are addressed: Does a given Lie group or Lie algebra act effectively on a given manifold? How smooth can such actions be? What fxed-point sets are possible? What happens under perturbations? Old results are summarized, and new ones presented, including: For every integer n there are solvable (in some cases, nilpotent) Lie algebras g that have effective C-infinity actions on all n-manifolds, but on some (in many cases, all) n-manifolds, g does not have effective analytic actions
Common zeros of inward vector fields on surfaces
Morris W. Hirsch
Mathematics , 2012,
Abstract: 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.
Chain transitive sets for smooth strongly monotone dynamical systems
Morris W. Hirsch
Mathematics , 2012,
Abstract: Let K denote a compact invariant set for a strongly monotone semiflow in an ordered Banach space E, satisfying standard smoothness and compactness assumptions. Suppose the semiflow restricted to K is chain transitive. The main result is that either K is unordered, or else K is contained in totally ordered, compact arc of stationary points; and the latter cannot occur if the semiflow is real analytic and dissipative. As an application, entropy is 0 when E = R^3 . Analogous results are proved for maps. The main tools are results of Mierczynski [27 ] and Terescak [37 ]
Smooth actions of Lie groups and Lie algebras on manifolds
Morris W. Hirsch
Mathematics , 2012, DOI: 10.1007/s11784-011-0069-5
Abstract: Necessary or sufficient conditions are presented for the existence of various types of actions of Lie groups and Lie algebras on manifolds.
Fixed points of local actions of nilpotent Lie groups on surfaces
Morris W. Hirsch
Mathematics , 2014,
Abstract: Let $G$ be connected nilpotent Lie group acting locally on a real surface $M$. Let $\varphi$ be the local flow on $M$ induced by a $1$-parameter subgroup. Assume $K$ is a compact set of fixed points of $\varphi$ and $U$ is a neighborhood of $K$ containing no other fixed points. Theorem: If the Dold fixed-point index of $\varphi_t|U$ is nonzero for sufficiently small $t>0$, then ${\rm Fix} (G) \cap K \ne \emptyset$.
Common zeroes of families of smooth vector fields on surfaces
Morris W. Hirsch
Mathematics , 2015,
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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