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Geodesic metrics on fractals and applications to heat kernel estimates

Science China Mathematics 2023年第5期66卷 907-934页

作者： Qingsong Gu Ka-Sing Lau Hua Qiu Huo-Jun Ruan Department of Mathematics Nanjing UniversityNanjing 210093China Department of Mathematics The Chinese University of Hong KongHong KongChina Department of Mathematics University of PittsburghPittsburghPA 15217USA School of Mathematical Sciences Zhejiang UniversityHangzhou 310027China

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 further assume that the weight functions g := ga are generated by“symmetric” weights a. Let M be the domain of a such that Dgadefines 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 ∈ *** 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.

关键词： Brownian motion heat kernel metric chain condition nested fractal quasisymmetry resistance metric Sierpinski carpet weight function

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HEAT KERNEL ESTIMATES ON JULIA SETS

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Acta Mathematica Scientia 2017年第5期37卷 1399-1414页

作者：杨甍 -Department of Mathematical Sciences Tsinghua University

We give heat kernel estimates on Julia sets J（f;） for quadratic polynomials f c（z） = z;+ c for c in the main cardioid or the ±1/k-bulbs where k ≥ 2. First we use external ray parametrization to construct a regular, strongly local and conservative Dirichlet form on Julia set. Then we show that this Dirichlet form is a resistance form and the corresponding resistance metric induces the same topology as Euclidean metric. Finally, we give heat kernel estimates under the resistance metric.

关键词： Julia sets heat kernel resistance metric

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