In this article the relationship between conditions of p-differentiability, p-holomorphycity, and existence derivative a function p-complex variable is considered. The general form p-holomorphic found. sufficient for p-analyticity local invertibility are obtained. open mapping theorem principle maximum norm uniqueness proved.

We consider the dynamics of expanding semigroups generated by finitely many rational maps on the Riemann sphere. We show that for an analytic family of such semigroups, the Bowen parameter function is real-analytic and plurisubharmonic. Combining this with a result obtained by the first author, we show that if for each semigroup of such an analytic family of expanding semigroups satisfies the o...