K. Simon, B. Solomyak, M. Urban\'nski
Parabolic Iterated Function Systems with Overlaps I
Preprint series: Mathematica Gottingensis

MSC:

58F12 Structure of attractors (and repellors)

Abstract: 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. 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.


Keywords: parabolic, iterated function systems, limit set, Hausdorff dimenson, overlaps, pressure.