Dimension rigidity for cookie-cutter Cantor sets

TL;DR

This paper establishes that two cookie-cutter Cantor sets with identical symbolic coding are differentiably equivalent precisely when they share the same Hausdorff dimension, linking geometric measure and differentiability.

Contribution

It provides a necessary and sufficient condition for differentiable equivalence of cookie-cutter Cantor sets based on their Hausdorff dimensions.

Findings

Differentiable equivalence requires equal Hausdorff dimensions.

Symbolic coding determines the structure of cookie-cutter Cantor sets.

Hausdorff dimension is a key invariant for differentiability.

Abstract

We show that two cookie-cutter Cantor sets with the same symbolic coding are differentiably equivalent if and only if their Hausdorff dimensions are equal.