Conformal dimension and its attainment on self-similar Laakso-type fractal spaces

TL;DR

This paper characterizes when the Ahlfors regular conformal dimension of symmetric Laakso-type fractal spaces is achieved, providing explicit metrics and new examples, thereby clarifying obstructions to attainment.

Contribution

It introduces a simple condition for attainment of conformal dimension and constructs explicit metrics on Laakso-type fractals.

Findings

Established a condition for conformal dimension attainment.

Constructed explicit metrics achieving the conformal dimension.

Provided new examples and clarified obstructions.

Abstract

A general construction of Laakso-type fractal spaces was recently introduced by the first two authors. In this paper, we establish a simple condition characterizing when the Ahlfors regular conformal dimension of a symmetric Laakso-type fractal space is attained. The attaining metrics are constructed explicitly. This gives new examples of attainment and clarifies the possible obstructions.