On the origin of escort distributions for q-entropies

Abstract: We present an argument about the origin of escort distributions used in conjunction with the q-entropy in non-additive thermo-statistics. The origin of the escort distributions is ascribed to the fact that the effective statistical description of the underlying system is provided by the measured Gromov-Hausdorff limit of a sequence of manifolds having a warped product metric structure. We interpret the entropic parameter \ q \ appearing in escort distributions as a "dimension" related to the fibration structure of the underlying phase spaces.