In the previous paper [9] and [10], we gave some classification theorems of periodic maps with isolated singular sets on compact surfaces. In this paper, we will give complete classification theorems of periodic maps on compact surfaces as similar formulae to those of [9] and [10].

We consider inclusions of type III factors arising from finite group actions. We show the relation between actions and the corresponding subfactors, and compute conjugacy invariants for the subfactors.

In the present paper we obtain a multiplier theorem for Fourier-Jacobi expansions in the space lip$(\gamma,p)$, $\gamma p>1$, which extends an earlier result of Trebels [11].