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Zygmund vector fields, Hilbert transform and Fourier coefficients in shear coordinates
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 135, Number 6, December 2013
- pp. 1559-1600
- 10.1353/ajm.2013.0056
- Article
- Additional Information
We parametrize the space ${\cal Z}$ 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.