Ergodic properties of p-adic transformations have been studiedwith respect to Haar measure. This paper extends the study of theseproperties to measures beyond Haar measure. Under these measures, coefficients do not appear in equal proportions. Adding a rationalnumber that is not an integer then takes likely strings of coefficientsto one of two unlikely strings of coefficients. It follows fromthis inequality that translation by a rational number other than aninteger is singular with respect to these measures. Conjugationgives similar results for multiplication by rational numbers.
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