# Are there any exact results for Hausdorff Measure?

The computation of the Hausdorff measure is extremely difficult due to the infinum appearing in its definition. This has made the calculation of the Hausdorff measure for nearly all fractals difficult or impossible. However, there are some good bounds, given by Baoguo Jia, for some exactly self-similar fractals such as the Siernpinski Triangle and Koch Curve.

One exception to this trend are the cantor sets. Their Hausdorff measure has been explicitly computed.

My question is this. Are there any other non-trivial fractals that have known exact Hausdorff measures? Could any of these other fractals be used to help aid in the calculation of other Hausdorff measures?

Edit: I'm still looking for answers, but now I know that techniques that can link the Hausdorff measure of specific fractals to more general ones will have very sophisticated arguments.

• The logarithmic capacity is related to the notion of Hausdorff measure and Hausdorff dimension. If $p(z)=a_{0}+...+z^{n}$ is a monic polynomial with complex coefficients, then the logarithmic capacity for the Julia set that corresponds to the polynomial $p$ is always $1$. – Joseph Van Name Apr 29 '15 at 15:26

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).