Regularity of the spectrum for expanding maps

In this short note, we propose to extend differentiability (with respect to a multidimensional parameter) of a normalized eigenfunction associated to the simple, dominating eigenvalue of the weighted transfer operator for a uniformly expanding map, to the whole discrete spectrum. We do so by studying directly the regularity (with respect to the parameter) of the resolvent operator, applying a general regularity result for fixed points of maps having loss of regularity.