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http -> https (the question has been bumped anyway)
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Martin Sleziak
Martin Sleziak
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There are higher-dimensional generalizations of the cantor sets, such as the Sierpinski triangle, and Sierpinski carpet, and there are some results, see e.g McMullens paper.

There is also a general result regarding iterated function systems (IFS), where the fractal is invariant under say $k$ similarity maps, and that the images if these do not overlap. Then, the dimension $s$ is given by the solution of $\sum_{i=1}^k c_k^s = 1$, and $c_i$ is the contraction factor of the $i$'th map. (This is mentioned on wikipedias page with fractals listed after Hausdorff dimensionwikipedias page with fractals listed after Hausdorff dimension).

The trouble is of course when we have overlap of the similarity maps, so some points are counted "twice" in some sense, which completely messes up everything.

I don't have a good explanation for why we cannot consider general non-overlapping affine maps, but it has to do with that magnification in certain directions affect the dimension more, in some sense (make a line segment twice as thick, and its still a line segment, but magnification in the other direction makes it longer).

There are higher-dimensional generalizations of the cantor sets, such as the Sierpinski triangle, and Sierpinski carpet, and there are some results, see e.g McMullens paper.

There is also a general result regarding iterated function systems (IFS), where the fractal is invariant under say $k$ similarity maps, and that the images if these do not overlap. Then, the dimension $s$ is given by the solution of $\sum_{i=1}^k c_k^s = 1$, and $c_i$ is the contraction factor of the $i$'th map. (This is mentioned on wikipedias page with fractals listed after Hausdorff dimension).

The trouble is of course when we have overlap of the similarity maps, so some points are counted "twice" in some sense, which completely messes up everything.

I don't have a good explanation for why we cannot consider general non-overlapping affine maps, but it has to do with that magnification in certain directions affect the dimension more, in some sense (make a line segment twice as thick, and its still a line segment, but magnification in the other direction makes it longer).

There are higher-dimensional generalizations of the cantor sets, such as the Sierpinski triangle, and Sierpinski carpet, and there are some results, see e.g McMullens paper.

There is also a general result regarding iterated function systems (IFS), where the fractal is invariant under say $k$ similarity maps, and that the images if these do not overlap. Then, the dimension $s$ is given by the solution of $\sum_{i=1}^k c_k^s = 1$, and $c_i$ is the contraction factor of the $i$'th map. (This is mentioned on wikipedias page with fractals listed after Hausdorff dimension).

The trouble is of course when we have overlap of the similarity maps, so some points are counted "twice" in some sense, which completely messes up everything.

I don't have a good explanation for why we cannot consider general non-overlapping affine maps, but it has to do with that magnification in certain directions affect the dimension more, in some sense (make a line segment twice as thick, and its still a line segment, but magnification in the other direction makes it longer).

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Per Alexandersson
Per Alexandersson
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There are higher-dimensional generalizations of the cantor sets, such as the Sierpinski triangle, and Sierpinski carpet, and there are some results, see e.g McMullens paper.

There is also a general result regarding iterated function systems (IFS), where the fractal is invariant under say $k$ similarity maps, and that the images if these do not overlap. Then, the dimension $s$ is given by the solution of $\sum_{i=1}^k c_k^s = 1$, and $c_i$ is the contraction factor of the $i$'th map. (This is mentioned on wikipedias page with fractals listed after Hausdorff dimension).

The trouble is of course when we have overlap of the similarity maps, so some points are counted "twice" in some sense, which completely messes up everything.

I don't have a good explanation for why we cannot consider general non-overlapping affine maps, but it has to do with that magnification in certain directions affect the dimension more, in some sense (make a line segment twice as thick, and its still a line segment, but magnification in the other direction makes it longer).