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It is well known that for a Brownian motion, if we change the medium to be inhomogeneous by a measure μ, then the new motion (the time-changed process) will diffuse according to a different metric D(·,·). In 2009, Kigami initiated a general scheme to construct such metrics through some self-similar weight functions gon the symbolic space. In order to provide concrete models to Kigami’s theoretical construction, in this paper, we give a thorough study of his metric on two classes of fractals of primary importance: the nested fractals and the generalized Sierpinski carpets; we assume further that the weight functions g≔ gaare generated by “symmetric” weights a. Let Mbe the domain of asuch that Dgadefines a metric, and let Sbe the boundary of M. One of our main results is that the metrics from gasatisfy the metric chain condition if and only if a∈ S.To determine Mand S, we provide a recursive weight transfer construction on the nested fractals, and a basic symmetric argument on the Sierpinski carpet. As an application, we use the metric chain condition to obtain the lower estimate of the sub-Gaussian heat kernel. This together with the upper estimate obtained by Kigami allows us to have a concrete class of metrics for the time change, and the two-sided sub-Gaussian heat kernel estimate on the fundamental fractals.

In the paper, we provide an effective criterion for the Lipschitz equivalence of two‐branch Cantor sets and three‐branch Cantor sets by studying the irreducibility of polynomials. We also find that any two Cantor sets are Lipschitz equivalent if and only if their contraction vectors are equivalent provided one of the contraction vectors is homogeneous.

Abstract: A self-similar measure on is defined to be a probability measure satisfyingwhere S j x = ρ j R j x + b j , T j x = η j Q j x + c j are contractive similarities, R j , Q j are orthogonal matrix and μ ∗ μ is the convolution of two measures. When M =0, μ is a linear self-similar measure, we establish the asymptotic behavior of averages of the derivative of the Fourier transform of μ, such asfor any order derivation of as R →∞ under certain additional hypotheses. When M >0, μ is a nonlinear self-similar measure, we get some results of L p boundedness for maximal operators of μ, from the pointwise asymptotic estimate of the Fourier transform of μ made by Strichartz. [Copyright &y& Elsevier]