We study parabolic iterated function systems with overlaps on the real line. We show that if a d-parameter family of such systems satisfies a transversality condition, then for almost every parameter value the Hausdorff dimension of the limit set is the minimum of 1 and the least zero of the pressure function. Moreover, the local dimension of the exceptional set of parameters is estimated. If the least zero is greater than 1, then the limit set (typically) has positive Lebesgue measure. These results are applied to some specific families including those arising from a class of continued fractions.
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