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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 g on 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 ≔ ga are generated by "symmetric" weights a. Let M be the domain of a such that Dga defines a metric, and let S be the boundary of M . One of our main results is that the metrics from ga satisfy the metric chain condition if and only if a ∈ S. To determine M and 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. [ABSTRACT FROM AUTHOR]

Although the 'hot spots' conjecture was proved to be false on some classical domains, the problem still generates a lot of interests on identifying the domains that the conjecture hold. The question can also be asked on fractal sets that admit Laplacians. It is known that the conjecture holds on the Sierpinski gasket and its variants. In this note, we show surprisingly that the 'hot spots' conjecture fails on the hexagasket, a typical nested fractal set. The technique we use is the spectral decimation method of eigenvalues of Laplacian on fractals. [ABSTRACT FROM AUTHOR]

In this paper, we first characterize the finiteness of fractal interpolation functions (FIFs) on post critical finite self-similar sets. Then we study the Laplacian of FIFs with uniform vertical scaling factors on the Sierpinski gasket (SG). As an application, we prove that the solution of the following Dirichlet problem on SG is a FIF with uniform vertical scaling factor 1/5: Δ u = 0 on SG { q , q , q }, and u( q ) = a , i = 1, 2, 3, where q , i = 1, 2, 3, are boundary points of SG. [ABSTRACT FROM AUTHOR]

This paper investigates the Lipschitz equivalence of generalized {1,3,5}-{1,4,5} self-similar sets D = ( r 1 D) ∪ ( r 2 D + (1 + r 1− r 2− r 3)/2) ∪ ( r 3 D + 1− r 3) and E = ( r 1 E) ∪ ( r 2 E + 1− r 2− r 3) ∪ ( r 3 E + 1− r 3), and proves that D and E are Lipschitz equivalent if and only if there are positive integers m and n such that r = r . [ABSTRACT FROM AUTHOR]

This paper deals with the Lipschitz equivalence of slidings of self-similar sets by graph-directed construction and martingale theory. [ABSTRACT FROM AUTHOR]