Narrow Results

Fractal subsets of ℝⁿ with a highly regular structure are often constructed as a limit of a recursive procedure based on contractive maps.

The Hausdorff dimension of recursively constructed fractals is relatively easy to find when the contractive maps associated with each recursive step satisfy the Open Set Condition (OSC). We present a class of random recursive constructions which resemble snowflake structures and which break the OSC. We calculate the associated Hausdorff dimension and conjecture that an a.s. deterministic exact Hausdorff function does not exist.

We prove that the modified von Koch snowflake curve, which we get as a limit by starting from an equilateral triangle (or from a segment) and repeatedly replacing the middle portion c of each interval by the other two sides of an equilateral triangle (and the corresponding von Koch snowflake domain), is non-self-intersecting if and only if c < ½. This answers a question of M. van den Berg.

Mendel and Naor's definition of metric cotype extends the notion of the Rademacher cotype of a Banach space to all metric spaces. Every Banach space has metric cotype at least 2. We show that any metric space that is bi-Lipschitz is equivalent to an ultrametric space having infimal metric cotype 1. We discuss the invariance of metric cotype inequalities under snowflaking mappings and Gromov–Hausdorff limits, and use these facts to establish a partial converse of the main result.