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Consider a graph directed iterated function system (GIFS) on the line which consists of similarities. Assuming neither any separation conditions, nor any restrictions on the contractions, we compute the almost sure dimension of the attractor. Then we apply our result to give a partial answer to an open problem in the field of fractal image recognition concerning some self-affine graph directed attractors in space.
This is an expository paper dealing with Jan Marik's results concerning perimeter and the divergence theorem of Gauss-Green-Ostrogradski.
For any , let
be its Lüroth expansion. Denote by the partial sum of the first terms in the above series and call it the th convergent of in the Lüroth expansion. This paper is concerned with the efficiency of approximating real numbers by their convergents in the Lüroth expansion. It is shown that almost no points can have convergents as the optimal approximation for infinitely many times in the Lüroth expansion. Consequently, Hausdorff dimension is introduced to quantify the set of...
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