Graphs that are not minimal for conformal dimension

TL;DR

This paper constructs functions with graphs of high Hausdorff dimension but low conformal dimension, providing counterexamples to previous assumptions and answering a specific open question in geometric analysis.

Contribution

It introduces functions whose graphs defy minimality of conformal dimension, expanding understanding of fractal geometry and quasisymmetric mappings.

Findings

Graphs with Hausdorff dimension > 1 but conformal dimension 1.

Existence of graphs with Hausdorff dimension 2 but conformal dimension 1.

Counterexamples to the minimality of conformal dimension for certain fractal graphs.

Abstract

We construct functions $f \colon [0,1] \to [0,1]$ whose graph as a subset of $\mathbb{R}^2$ has Hausdorff dimension greater than any given value $\alpha \in (1,2)$ but conformal dimension $1$. These functions have the property that a positive proportion of level sets have positive codimension-$1$ measure. This result gives a negative answer to a question of Binder--Hakobyan--Li. We also give a function whose graph has Hausdorff dimension $2$ but conformal dimension $1$. The construction is based on the author's previous solution to the inverse absolute continuity problem for quasisymmetric mappings.