Analysis on fractals, Laplacians on self-similar sets, noncommutative geometry and spectral dimensions
From author's introduction: ``In this paper, we address the question of finding a suitable analogue of Weyl's asymptotic formula for the eigenvalue distribution of Laplacians on (certain classes of self- similar) fractals. We propose, in particular, an analogue of the notion of Riemannian volume on fractals and establish, in the process, some connections between analysis on fractals, spectral geometry, and aspects of Connes' noncommutative geometry. Several questions arise naturally in this context: \(Q_0)\) What is a (self-similar) fractal? \(Q_1)\) What is a Laplacians on a fractal? \(Q_2)\) Is there an analogue of Weyl's formula for Laplacian on a fractal \(F\)? If so, what is its form? \(Q_3)\) What is the value of the ``spectral exponent'' \(d_s\)? Further, what is the geometrical (or analytical) interpretation of \(d_s\)? \(Q_4)\) Is there an analogue of the notion of ``Riemannian volume'' on a fractal? \(Q_5)\) What is a suitable analogue of the notion of ``geodesic flow'' on a fractal? In this paper we shall address or revisit several of these questions and propose possible answers and/or conjectures. Our goal is to help develop further (geometric) analysis on ``fractals'', as well as to establish further connections between this exciting new subject and aspects of Connes' noncommutative geometry''.
scientific article; zbMATH DE number 733361