Morse Theory on Spaces of Braids and Lagrangian Dynamics
Author(s)
Ghrist, Robert W.
Van den Berg, J. B.
Vandervorst, R. C.
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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 and provides a gloablization of 'lap number' techniques used in scalar parabolic PDEs.
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.
Sponsor
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.
Date
2001
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