I am looking to find the Hausdorff dimension of the following object. It is composed of self-similar disjoint annuli. This is my current work.
If anybody can help describe the steps to find the Hausdorff dimension of this object I would really appreciate it. Thank you.
Dilating $\Omega$ by a factor of $\frac14$ yields a copy, say $\Omega'$, of $\Omega$. Because of that, The area of $\Omega'$, $A(\Omega') = \frac{1}{16} A(\Omega)$.
Thus, the scaling factor is $\frac{1}{16}$. The Hausdorff dimension $D$ of $\Omega$ is:
$$\frac{1}{16} = \Big(\frac{1}{4}\Big)^D \Leftrightarrow D=2.$$