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The zeta function of the Laplacian on certain fractals. (English) Zbl 1136.30002

Diffusion on fractals can be seen as a generalisation of usual Brownian motion on manifolds. The Laplacian can be also seen as the infinitesimal generator of a Brownian motion.
The authors introduce the zeta function \[ \zeta_{\Delta} (s)=\sum_{-\Delta u =\mu u} \frac{1}{\mu^{s}} \] of the Laplacian \(\Delta\) on self-similar fractals, and obtain properties of the poles, residues, counting function of \(\zeta_{\Delta}(s)\), etc.

MSC:

30B50 Dirichlet series, exponential series and other series in one complex variable
11M41 Other Dirichlet series and zeta functions
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
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