Reverse Lipschitz surjections without commensurability

Determine whether, for dust-like self-similar sets of equal Hausdorff dimension, the existence of a Lipschitz surjection from one set onto the other implies the existence of a Lipschitz surjection in the reverse direction without assuming that either set has commensurable similarity ratios.

Background

The paper proves that if two dust-like self-similar sets have equal Hausdorff dimension, at least one of them has commensurable similarity ratios, and there is a Lipschitz surjection from X onto Y, then a Lipschitz surjection also exists from Y onto X. The result follows from the paper’s symmetric characterization of Lipschitz surjections under the commensurability assumption.

The unresolved issue is whether this symmetry of Lipschitz surjections persists for arbitrary dust-like self-similar sets of equal dimension when neither set is assumed to have commensurable ratios. The question concerns the limitations of the commensurability hypothesis in the reverse-surjection corollary and is not resolved by the results established in the paper.