Morse theory on spaces of braids and Lagrangian dynamics |
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Authors: | RW Ghrist JB Van den Berg RC Vandervorst |
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Institution: | (1) Department of Mathematics, University of Illinois, Urbana, IL 61801, USA, US;(2) Department of Applied Mathematics, University of Nottingham, UK, GB;(3) Department of Mathematics, Free University Amsterdam, De Boelelaan 1081, Amsterdam, Netherlands, NL;(4) CDSNS, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA, US |
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Abstract: | In the first half of the paper we construct a Morse-type theory on certain spaces of braid diagrams. We define a topological
invariant of closed positive braids which is correlated with the existence of invariant sets of parabolic flows defined on discretized braid spaces. Parabolic flows, a type of one-dimensional lattice dynamics, evolve singular braid diagrams
in such a way as to decrease their topological complexity; algebraic lengths decrease monotonically. This topological invariant
is derived from a Morse-Conley homotopy index.?In the second half of the paper we apply this technology to second order Lagrangians
via a discrete formulation of the variational problem. This culminates in a very general forcing theorem for the existence
of infinitely many braid classes of closed orbits.
Oblatum 11-V-2001 & 13-XI-2002?Published online: 24 February 2003
RID="*"
ID="*"The first author was supported by NSF DMS-9971629 and NSF DMS-0134408. The second author was supported by an EPSRC Fellowship.
The third author was supported by NWO Vidi-grant 639.032.202. |
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