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Appears in collection : French Japanese Conference on Probability and Interactions

This talk will present the speaker's ongoing work on “geometrically canonical” Laplacians on self-conformal fractal curves in the plane. The main result is that on a given such curve one can construct a family of Laplacians whose heat kernels and eigenvalue asymptotics “respect” the fractal nature of the Euclidean geometry of the curve in certain nice ways. The idea of the construction of such Laplacians originated from the speaker's preceding studies on the case of a circle packing fractal, i.e., a fractal subset of $\mathbb{C}$ whose Lebesgue area is zero and whose complement in the Riemann sphere $\widehat{\mathbb{C}}:=\mathbb{C}\cup{\infty}$ is the union of disjoint open disks in $\widehat{\mathbb{C}}$. He has observed that, on a given such fractal, one can explicitly define a Dirichlet form (a quadratic energy functional) by a certain weighted sum of the standard one-dimensional Dirichlet form on each of the circles constituting the fractal, and that this Dirichlet form “respect” the Euclidean geometry of the fractal in the sense that the inclusion map of the fractal into $\mathbb{C}$ is harmonic with respect to this form. The speaker has also proved that such a Dirichlet form is unique for the classical Apollonian gaskets and that, for some concrete families of self-conformal circle packing fractals including the Apollonian gaskets, the associated Laplacian satisfies Weyl's eigenvalue asymptotics involving the Euclidean Hausdorff dimension and measure of the fractal. It would be desirable if one could extend such results to self-conformal fractals which are not circle packing ones, and the talk will present an extension to the simplest case of self-conformal fractal curves in the plane. The key point of the construction of Laplacians is to use (suitable versions of) the harmonic measure in defining the Dirichlet form BUT to use fractional-order Besov seminorms (with respect to the harmonic measure) of the inclusion map into $\mathbb{C}$ in defining the $L^{2}$-inner product for functions on the fractal.

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