In modern study of geometry of attractors of iterated function systems, one tries to say something about the geometry of K while assuming as little of the functions f_i or the placement of the sets f_i(K) as possible. For example, the functions f_i may be assumed to be only affine, or even merely continuously differentiable, and already in many works there are no assumptions made at all regarding pairwise intersections of f_i(K). One of the driving questions in this field is the following: To what extent is the geometry of K determined by algebraic properties of the semigroup generated by (f_1, ..., f_n) under composition? In principle, this semigroup should capture all the geometric data of K, including its Hausdorff dimension. As a concrete question, I will discuss how any "arithmetic structure" present in the semigroup generated by (f_1, ..., f_n) turns out to manifest as "arithmetic structure" on the attractor K.
28.2. Aleksi Pyörälä: On geometry of attractors of iterated function systems
Abstract: Given a finite collection of distance-contracting functions f_1, ..., f_n on the plane, a compact set K in the plane is called the attractor of the iterated function system (f_1, ..., f_n) if K = \bigcup_{i=1}^n f_i(K). In the special case when all the functions f_1, ..., f_n are similarities, the set K is called self-similar. Perhaps the most classical examples of self-similar sets in the plane are the Sierpinski carpet and the four-corner Cantor set. Indeed, the classical approach in studying the geometry of self-similar sets is to make the assumption that the pieces f_i(K) the set K consists of are far from each other, or at least that they intersect only in very minor ways. Such an assumption forces the whole set K to be "well-spaced" in the plane: In particular, K will be Ahlfors regular. By now, the geometry of self-similar sets with such separation assumptions on the sets f_i(K) is quite well understood.
