Consider a random linear iterated function system on the line, with fixed contractions and random translations. This type of systems was studied, with Jordan, Pollicott, and Simon proving that when the similarity dimension $s<1$ then the Hausdorff dimension of the attractor is almost surely equal to $s$, and if $s>1$ then the attractor has almost surely positive Lebesgue measure. The latter case was recently improved by Dekking, Simon, Szekely, and Szekeres, who proved that if $s>1$ then the attractor almost surely contains an open interval.

In this talk I will present our latest work with Balazs Barany, in which we prove that actually, if $s>1$ and under some non-restrictive conditions on the probability distribution, the density of the (random) natural measure is Holder continuous. The proof uses an approach we learned from a paper of Erraoui and Hakiki on fractional Brownian motions.

Meeting ID: 852 4277 3200 Passcode: 103121