Families of infinite parabolic IFS with overlaps the approximating method

This work is devoted to the study of families of infinite parabolic iterated function systems (PIFS) on a closed interval $X\subset\mathbb{R}$ parametrized by $\bold{t}\in U\subset \mathbb{R}^d$ with overlaps. We show that the Hausdorff dimension and absolute continuity of ergodic projections through the families of infinite PIFS are decided \emph{a.e.} by the growth rate of the entropy of the sequence of concentrating measures and Lyapunov exponents of the family of truncated PIFS, under transversality of the families essentially. We also give an estimation on the upper bound of the Hausdorff dimension of parameters where the corresponding ergodic projections admit certain dimension drop. The setwise topology on the space of measures enables us to approximate the families of the infinite systems by families of its finite truncated sub-systems, which plays the key role throughout our work.