On the shape of Cantor sets (Q1263852)

Sets of self-similar, fractal nature have been studied extensively recently. As a prototype of a more general problem, the authors study a class of Cantor sets of the line from the point of view of bi-Lipschitz geometry, i.e. quasi-isometry. These investigations reveal a general principle: A quasi-isometry between such objects is essentially the same thing as a map which is linear on the level of measure theory, i.e., has constant Radon-Nikodým derivative with respect to Hausdorff measure. This principle provides new invariants which enable the authors to classify generic Cantor sets of this type.