Timeline for Iterated function system on the plane

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Jul 10, 2013 at 18:55	comment	added	Pablo Shmerkin		In fact, if the similarities are homotheties, then open set condition CANNOT hold if $r_i+r_j>1$ for some $1\le i<j\le 3$. This is because for an IFS consisting of two planar homotheties $f_i,f_j$ with ratios $r_i,r_j$, the attractor is the (possible degenerate) segment joining the fixed points and thus has Hausdorff dimension $1$ (or $0$) which is strictly smaller than the similarity dimension, so the open set condition cannot hold. So much less can it hold if you add yet another map.
Jul 10, 2013 at 18:51	comment	added	Pablo Shmerkin		There are multiple issues with this answer. Falconer's theorem requires the norm of the maps to be $<1/2$, in this case this means $r_1,r_2,r_3<1/2$ in which case the result is trivial. In fact for this family of IFS's, the fixed points $p_1,p_2,p_3$ play no role, as long as they're in general position the self-similar sets obtained are all affine images of each other. It is also most definitely not true that the open set condition holds if the Hausdorff dimension equals the similarity dimension (only the converse is true), so there is no reason to believe the pieces $f_i(K)$ will be disjoint.
Jul 10, 2013 at 16:24	history	edited	Gerald Edgar	CC BY-SA 3.0	deleted 5 characters in body
Jul 10, 2013 at 16:18	history	answered	Gerald Edgar	CC BY-SA 3.0