On the triangular self-affine iterated function systems

Michał Rams (IMPAN)

Frank Adams 1,

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.

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