A pull-back measure formula obtained in some particular cases by E. A. Nordgren and this author is generalized in the framework of boundary measures for zero-free Nevanlinna class fuctions on the unit polydisk. The formula is used to characterize the zero-free Nevanlinna class functions which are solutions of Schröder's equation induced by a polydisk automorphism ϕ (i.e. to determine the zero-free functionsf belonging to the Nevanlinna class which are solutions of the functional equationf ° π=λf, for some constant λ), thus generalizing earlier results obtained by R. Mortini and this author.