Hutchinson's formula for asymptotically homogeneous Cantor sets - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T06:08:16Z http://mathoverflow.net/feeds/question/118811 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/118811/hutchinsons-formula-for-asymptotically-homogeneous-cantor-sets Hutchinson's formula for asymptotically homogeneous Cantor sets Nikita Sidorov 2013-01-13T14:10:21Z 2013-01-13T17:00:25Z <p>As everyone knows, the standard middle-thirds Cantor set is constructed by dividing the interval into three equal parts, removing the middle one, then applying the same procedure to the remaining two intervals, etc. </p> <p>The resulting set has Hausdorff dimension $s=\log 2/\log 3$, in view of Hutchinson's formula: $$\sum_{i=1}^m r_i^s=1,$$ where in our case $m=2, r_1=r_2=1/3$. </p> <p>Now, assume that $m=2$ but that on level $n$ we remove the middle-thirds interval whose relative measure is not exactly $1/3$ but $1/3+\delta_n$, where $\delta_n\to0$ sufficiently fast. (It may also depend on a position of the interval but there is always a uniform upper bound which tends to 0 sufficiently fast.) </p> <p>Is it still true that the Hausdorff dimension of such a set is $\log2/\log3$? </p> <p>This is actually a `toy question', since in my set-up the corresponding iterated function system is infinite countable. However, Hutchinson's formula works for such IFS just as well (with $m=\infty$), so I'm sure the conclusion should be the same. If it helps, the uniform upper bound for the $\delta_n$ in my case is a double exponent, i.e., $\frac1n \log (-\log \delta_n)\to \text{const}$ as $n\to+\infty$. </p> http://mathoverflow.net/questions/118811/hutchinsons-formula-for-asymptotically-homogeneous-cantor-sets/118818#118818 Answer by R W for Hutchinson's formula for asymptotically homogeneous Cantor sets R W 2013-01-13T15:33:39Z 2013-01-13T15:33:39Z <p>Since you remove "more", it should be more or less clear that $\text{HD}\; C_\delta \le \text{HD}\; C$ (where $C$ is the standard Cantor set and $C_\delta$ is the "perturbation" you describe). For proving the opposite inequality it is enough to exhibit a measure on $C_\delta$ whose Hausdorff dimension is equal to $\text{HD}\; C$. This is the usual uniform measure $m$ on $C_\delta$ (i.e., the one whose value is $1/2^n$ on each of the rank $n$ intervals), because if $\delta_n\to 0$ fast enough, then $$\frac{\log m B(x,r)}{\log r} \to \log 2/\log 3$$ for $m$-a.e. $x\in C_\delta$.</p> http://mathoverflow.net/questions/118811/hutchinsons-formula-for-asymptotically-homogeneous-cantor-sets/118826#118826 Answer by BSteinhurst for Hutchinson's formula for asymptotically homogeneous Cantor sets BSteinhurst 2013-01-13T17:00:25Z 2013-01-13T17:00:25Z <p>Just a quick note before having to move on, but when thinking about IFSs which approached self-similarity the papers of Igudesman have been useful. He has a notion of lacunary self-similar sets which (without checking details) looks like your set up. The main paper I am thinking of is the one referred to <a href="http://www.emis.de/journals/LJM/content12.htm" rel="nofollow">here</a>.</p>