  We exhibit the duality between best Lipschitz (infinity harmonic) maps and
least gradient maps in the case of maps from surfaces to the circle. We show
that given a homotopy class of a map from a surface to the circle the infinity
harmonic map defines a geodesic lamination on the surface and the dual least
gradient map defines a transverse measure on the lamination. This is the
initial step towards an analytic approach to Thurston's work on best Lipschitz
maps between hyperbolic surfaces and Thurston's asymmetric metric on
Teichmueller space.
