We study the relationship between the Assouad dimension and quasisymmetric mappings, showing that spaces of dimension strictly less than one can be quasisymmetrically deformed onto spaces of arbitrarily small dimension. We conjecture that this fact holds also for the Hausdorff dimension, and our results yield several corollaries which provide partial support for this conjecture. The proofs make use of connections between Assouad dimension, porosity, and ultrametrics.
"Lowering the Assouad dimension by quasisymmetric mappings." Illinois J. Math. 45 (2) 641 - 656, Summer 2001. https://doi.org/10.1215/ijm/1258138361