, Volume 152, Issue 2, pp 369-432

Morse theory on spaces of braids and Lagrangian dynamics

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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
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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.