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This paper deals with both complex dynamical systems and conformal iteratedfunction systems. We study finitely generated expanding semigroups of rationalmaps with overlaps on the Riemann sphere. We show that if a $d$-parameterfamily of such semigroups satisfies the transversality condition, then foralmost every parameter value the Hausdorff dimension of the Julia set is theminimum of 2 and the zero of the pressure function. Moreover, the Hausdorffdimension of the exceptional set of parameters is estimated. We also show thatif the zero of the pressure function is greater than 2, then typically the2-dimensional Lebesgue measure of the Julia set is positive. Some sufficientconditions for a family to satisfy the transversality conditions are given. Wegive non-trivial examples of families of semigroups of non-linear polynomialswith transversality condition for which the Hausdorff dimension of the Juliaset is typically equal to the zero of the pressure function and is less than 2.We also show that a family of small perturbations of Sierpi\'nski gasket systemsatisfies that for a typical parameter value, the Hausdorff dimension of theJulia set (limit set) is equal to the zero of the pressure function, which isequal to the similarity dimension. Combining the arguments on thetransversality condition, thermodynamical formalisms and potential theory, weshow that for each complex number $a$ with $|a|\neq 0,1$, the family of smallperturbations of the semigroup generated by ${z^{2}, az^2} $ satisfies that fora typical parameter value, the 2-dimensional Lebesgue measure of the Julia setis positive.

Transversality Family of Expanding Rational Semigroups