**MSC:**- 58F12 Structure of attractors (and repellors)

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. 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 one arising from a class of continued fractions.