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Self-affine and self-similar sets: measures and intersections

Prof. Tamás Keleti
Eötvös Loránd University, Budapest

21 - 9 - 2005

Let $K$ be a self-affine set in ${\mathbb{R}}^n$ and let $m$ be a "natural" probability measure on $K$. We study the following 3 types of questions:

1. Does there exist a $C<1$ (for $K$ and $m$) such that for any affine map / similarity / isometry / translation $f$
   
   $$\mbox{\textlatin {either}} \quad m(K\cap f(K)) \le C \qquad \mbox{\textlatin {or}} \quad f(K)\supset K?$$
   
   

2. Is it true that for any affine map / similarity / isometry / translation $f$, $m(K\cap f(K)) > 0$ iff $f(K)$ has non-empty interior in $K$ (that is, $f(K)$ contains an elementary part of $K$)?

3. Can at least one / most (unless there is some clear obstacle) $m$ can be extended to an isometry / translation invariant Borel measure $b$ on ${\mathbb{R}}^n$ such that $b(K)=1$?

We have some positive results among others if $K$ is a self-similar set with B disjoint parts and also if $K$ is self-affine sponge (we get by subdividing a cube into $k_1 \times k_2 \times \ldots \times k_n$ parts uniformly and taking some of them and then iterating this).

joint work with Márton Elekes and András Máthé