Peter Varju (University of Cambridge)
Title:
On the dimension of self-similar measures
Abstract:
Let f_1,...,f_n be a collection of contracting similarities on R, and let p_1,...,p_n be a probability vector. There is a unique probability measure mu on R that satisfies the identity
mu = p_1 f_1(mu) + ... + p_n f_n(mu).
This measure is called self-similar. The maps f_1,...,f_n are said to satisfy the no exact overlaps condition if they generate a free semigroup (i.e. all compositions are distinct). Under this condition, the dimension of mu is conjectured to be the minimum of 1 and the ratio of the entropy of p_1,...,p_n and the average logarithmic contraction factor of the f_i. This conjecture has been recently established in some special cases, including when n=2 and f_1 and f_2 have the same contraction factor. In the talk I will discuss recent progress by Ariel Rapaport and myself in the case n=3. In this case new difficulties arise as was demonstrated by recent examples of Baker and Barany, Kaenmaki of IFS's with arbitrarily weak separation properties.