Summary: Appendix to A rigidity theorem for the solvable
August 18, 1997
The study of properties of a metric space which are preserved by bilipschitz homeomorphism occurs
in the study of groups via a word metric. It was also studied in  for a certain type of Cantor
set embedded in the real line. The Cantor sets concerned are mild generalizations of the original
middlethird Cantor set. This Cantor set has the basic property that it is the union of two exact
copies of itself each scaled down in size by a factor of 1=3: The generalization allows finitely many
linear scale factors. It is easy to see that the Hausdorff dimension of this type of Cantor set depends
only on these scale factors. Now a bilipschitz homeomorphism preserves Hausdorff dimension, and
so a natural question is what, if any, further invariants other than Hausdorff dimension are there.
An almost complete answer was given in , using invariants derived from the Hausdorff measure.
This work was generalized to analogous Cantor sets in Euclidean space of dimension n by H. Vuong
in his thesis, , . In this section, we generalize in a different direction to abstract metric Cantor
sets which possess a certain linear selfsimilarity structure. There is a further generalization to a
much wider class of Cantor set, where the selfsimilarity structure is smooth rather than linear. This
will not be dealt with here.
The main result is Theorem (0.6) which states that every bilipschitz homeomorphism between