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

2022/11/21 by Sasaya, Kôhei
#30L10 #31E05 #35K08 #60J35 #FOS: Mathematics #Metric Geometry (math.MG) #Probability (math.PR)

paper · doi:10.48550/arxiv.2211.11473

Abstract

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 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_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.

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