  Let $\{a_n(x)\}_{n\geq1}$ be the sequence of digits of $x\in(0,1)$ in
infinite iterated function systems with polynomial decay of the derivative. We
first study the multifractal spectrum of the convergence exponent defined by
the sequence of the digits $\{a_n(x)\}_{n\geq1}$ and the weighted products of
distinct digits with finite numbers respectively, and then calculate the
Hausdorff dimensions of the intersection of sets defined by the convergence
exponent of the weighted product of distinct digits with finite numbers and
sets of points whose digits are non-decreasing in such iterated function
systems.
