  We parametrize the space $\mathcal{Z}$ of Zygmund vector fields on the unit
circle in terms of infinitesimal shear functions on the Farey tesselation. Then
we express the Hilbert transform and the Fourier coefficients of the Zygmund
vector fields in terms of the above parametrization by infinitesimal shear
functions. Finally, we compute the Weil-Petersson metric on the Teichm\"uller
space of a punctured surface in terms of shears.

Geometric Topology;Complex Variables