relevant operators from a world-sheet Renormalization-Group viewpoint. One distinguishes two cases. In the first, the initial RW spacetime does not possess cosmological horizons. In this case it is shown that the limit to the conformal world-sheet non-trivial (infrared) fixed point can be taken smoothly without problems. On the other hand, in the case where the initial spacetime has cosmological horizons, such a limit is plagued by world-sheet divergences. These should be carefully subtracted in order to allow for a smooth approach to the fixed point. A detailed discussion of how this can be done is presented. In general, the divergences spoil the conformal invariance of the $\sigma$-model, thus implying the need for Liouville dressing [@ddk] in order to properly restore the conformal symmetry.

Moreover, a careful discussion of the matching between the results of the Liouville dressing and those implied by the logarithmic algebra is given, which supports the possibility of identifying the world-sheet zero mode of the Liouville field (viewed as a local renormalization-group scale on the world sheet) with the target time. One distinguishes various cases which depend on whether the underlying theory lives in its critical dimension, and thus the only source of not criticality is the 