Do Mobius transformations preserve Hausdorff dimension?
This may be related to: Is there a measure invariant with respect to the Möbius transformation?
I believe the answer is yes, but I want some intuition behind it, and eventually, a proof.
Since a Mobius transformation is a combination of translations, dilations, rotations, and inversions, it suffices to show that each of these mappings preserves the Hausdorff dimension.
Motivation:
This image
and this image
are the same (up to an inversion/Mobius transformation). In both images, the "dust" or "residual set" left between the tangent circles has the same Hausdorff dimension $ \delta \approx 1.30568$.