Some inequalities between Ahlfors regular conformal dimension and spectral dimensions for resistance forms

TL;DR

This paper explores inequalities between the Ahlfors regular conformal dimension and spectral dimensions for resistance forms, establishing bounds and providing examples of their relationships on metric spaces.

Contribution

It proves an upper bound for the Ahlfors regular conformal dimension in terms of a variation of spectral dimension and constructs an example showing the opposite inequality.

Findings

Proved $ ext{dim}_ ext{ARC}(X,R) \u2264 ar{d}_s < 2$ for resistance forms.

Constructed an example with $d_s < ext{dim}_ ext{ARC}(X,R) < 2$.

Established inequalities linking conformal and spectral dimensions for resistance forms.

Abstract

Quasisymmetric maps are well-studied homeomorphisms between metric spaces preserving annuli, and the Ahlfors regular conformal dimension $\dim_\mathrm{ARC}(X,d)$ of a metric space $(X,d)$ is the infimum over the Hausdorff dimensions of the Ahlfors regular images of the space by quasisymmetric transformations. For a given regular Dirichlet form with the heat kernel, the spectral dimension $d_s$ is an exponent which indicates the short-time asymptotic behavior of the on-diagonal part of the heat kernel. In this paper, we consider the Dirichlet form induced by a resistance form on a set $X$ and the associated resistance metric $R$. We prove $\dim_\mathrm{ARC}(X,R)\le \overline{d_s}<2$ for $\overline{d_s}$, a variation of $d_s$ defined through the on-diagonal asymptotics of the heat kernel. We also give an example of a resistance form whose spectral dimension $d_s$ satisfies the opposite inequality $d_s<\dim_\mathrm{ARC}(X,R)<2.$