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

Dimensions of Julia sets of expanding rational semigroups