  We estimate the upper box and Hausdorff dimensions of the Julia set of an
expanding semigroup generated by finitely many rational functions, using the
thermodynamic formalism in ergodic theory. Furthermore, we show Bowen's
formula, and the existence and uniqueness of a conformal measure, for a
finitely generated expanding semigroup satisfying the open set condition. What
we discuss is the `backward iterated function systems', and since the generator
maps are not injective in general, we need some efforts to estimate the measure
of disks.
