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On the geometric structure of the limit set of conformal iterated function systems. (English) Zbl 1036.28007

Summary: We consider infinite conformal function systems on \(\mathbb{R}^d\). We study the geometric structure of the limit set of such systems. Suppose this limit set intersects some \(l\)-dimensional \(C^1\)-submanifold with positive Hausdorff \(t\)-dimensional measure, where \(0< l< d\) and \(t\) is the Hausdorff dimension of the limit set. We then show that the closure of the limit set belongs to some \(l\)-dimensional affine subspace or geometric sphere whenever \(d\) exceeds 2 and an analytic curve if \(d\) equals 2.

MSC:

28A80 Fractals
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