For any $q\in (0,1)$, one can consider a geometric series
+1 +q +q^2 +…,
and then toss a coin countably many times to decide whether each sign « + » is kept or is replaced by a minus one. The law of this random variable is given by the stationary measure for the random dynamical system, consisting of two affine maps
x\mapsto \pm 1 + qx,
taken with the probability (1/2) each. This stationary measure is called the Bernoulli convolution measure. It is supported on a Cantor set for $q\in (0,1/2)$, and on an interval for $q\in [1/2,1)$. Its properties — and most importantly, whether it is absolutely continuous or signular — have been studied for many years with many famous works and important recent progress in the domain (Erdos, Solomyak, Shmerkin, Varju, …).
My talk will be devoted to our recent work with P. Vytnova and M. Pollicott (https://arxiv.org/abs/2102.07714). I will present a technique for obtaining a lower bound for the Hausdorff dimension for the stationary measure of an affine IFS with similarities (in particular, affine IFS on the real line).