## On the triangular self-affine iterated function systems

#### Michał Rams (IMPAN)

The iterated function system is a finite family $$\{f_i\}$$ of uniformly contracting maps in $$\mathbb{R}^n$$. The limit set of an IFS is the unique nonempty compact set $$\Lambda$$ satisfying $$\Lambda = \bigcup f_i(\Lambda)$$. We are interested in providing a geometrical description (usually in the terms of Hausdorff dimension) of $$\Lambda$$ for large classes of IFSs.
In this talk I will present results (joint with Balazs Barany and Karoly Simon) on the Hausdorff dimension of the limit sets for (some class of) IFS for which the maps $$f_i: \mathbb{R}^2\to\mathbb{R}^2$$ are of the form $$f_i (x) = A_i x + a_i$$, where $$A_i$$ are uppertriangular matrices and $$a_i$$ are some translations. What's interesting, while one of our previous results (partially) answers this question for diagonal matrices, the result I will be speaking about not only is not a generalization of the previous one, but our method of proof does not work for diagonal matrices at all - among other assumptions, we have to assume that at least one of $$A_i$$'s has a nonzero nondiagonal element.