On the dimension of self-similar measures

Abstract: Let $f_1$,…,$f_n$ be a collection of contracting similarities on $\mathbb{R}$, and let $p_1$,…,$p_n$ be a probability vector. There is a unique probability measure mu on $\mathbb{R}$ 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.

dynamical systemsnumber theory

Audience: general audience

New England Dynamics and Number Theory Seminar