The UV and Ly$\alpha$ Luminosity Functions of galaxies and the Star
  Formation Rate Density at the end of HI reionization from the VIMOS
  Ultra-Deep Survey (VUDS)
  We establish a robust statistical description of the star-forming galaxy
population at the end of cosmic HI reionization ($5.0\le{}z\le6.6$) from a
large sample of 52 galaxies with spectroscopically confirmed redshifts from the
VIMOS UltraDeep Survey. We identify galaxies with Ly$\alpha$ either in
absorption or in emission, at variance with most spectroscopic samples in the
literature where Ly$\alpha$ emitters dominate. We find that star-forming
galaxies at these redshifts are distributed along a main sequence in the
stellar mass vs. SFR plane. We report a flat evolution of the sSFR(z) in 3<z<6
compared to lower redshift measurements. UV-continuum slopes vary with
luminosity, with a large dispersion. We determing UV and Ly$\alpha$ luminosity
functions using V$_{max}$ method and use them to derive star formation rate
densities (SFRD). We find that both UV-derived and Ly$\alpha$-derived SFRDs are
in excellent agreement after correcting Ly$\alpha$ luminosity density for IGM
absorption. Our new SFRD measurements at a mean redshift z=5.6 confirm the
steep decline of the SFRD at z>2. The bright end of the Ly$\alpha$ luminosity
function has a high number density, indicating a significant star formation
activity concentrated in the brightest Ly$\alpha$ emitters (LAE) at these
redshifts. LAE with EW>25\AA ~contribute to about 75\% of the total UV-derived
SFRD. While our analysis favors a low dust content in 5.0<z<6.6, uncertainties
on the dust extinction correction and associated degeneracies in spectral
fitting will remain an issue to estimate the total SFRD until future survey
extending spectroscopy to the NIR rest-frame spectral domain, e.g. with JWST.
