We present the results obtained with VLT/MUSE on the faint-end of the Lyman-alpha luminosity function (LF) based on deep observations of four lensing clusters. The precise aim of the present study is to further constrain the abundance of Lyman-alpha emitters (LAEs) by taking advantage of the magnification provided by lensing clusters. We blindly selected a sample of 156 LAEs, with redshifts between $2.9 le z le 6.7$ and magnification-corrected luminosities in the range $ 39 lesssim log L_{Ly_{alpha}}$ [erg s$^{-1}$] $lesssim 43$. The price to pay to benefit from magnification is a reduction of the effective volume of the survey, together with a more complex analysis procedure. To properly take into account the individual differences in detection conditions (including lensing configurations, spatial and spectral morphologies) when computing the LF, a new method based on the 1/Vmax approach was implemented. The LAE LF has been obtained in four different redshift bins with constraints down to $log L_{Ly_{alpha}} = 40.5$. From our data only, no significant evolution of LF mean slope can be found. When performing a Schechter analysis including data from the literature to complete the present sample a steep faint-end slope was measured varying from $alpha = -1.69^{+0.08}_{-0.08}$ to $alpha = -1.87^{+0.12}_{-0.12}$ between the lowest and the highest redshift bins. The contribution of the LAE population to the star formation rate density at $z sim 6$ is $lesssim 50$% depending on the luminosity limit considered, which is of the same order as the Lyman-break galaxy (LBG) contribution. The evolution of the LAE contribution with redshift depends on the assumed escape fraction of Lyman-alpha photons, and appears to slightly increase with increasing redshift when this fraction is conservatively set to one. (abridged)
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