Some Inequalities Between Ahlfors Regular Conformal Dimension And Spectral Dimensions For Resistance Forms.

Quasisymmetric maps are well-studied homeomorphisms between metric spaces preserving annuli, and the Ahlfors regular conformal dimension dim 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 that 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 ARC (X , R) ≤ d s ¯ < 2 for 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 ARC (X , R) < 2. [ABSTRACT FROM AUTHOR]