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Fractals and the base eigenvalue of the Laplacian on certain noncompact surfaces. (English) Zbl 1153.30035

Consider the group \(\Gamma=\Gamma_{\theta}\) (where \(\theta\in(0,\pi/2)\)) of Möbius transformations of the unit disk \(\mathbb{D}\) generated by
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reflection in the imaginary axis,
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reflection in the hyperbolic geodesic from \(i\) to \(e^{i\theta}\) and
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reflection in the hyperbolic geodesic from \(-i\) to \(e^{-i\theta}\).
The quotient \(\mathbb{D}/\Gamma\) is a hyperbolic surface; the author is interested in the smallest real eigenvalue \(\lambda_0\) of the Laplacian on this surface. This quantity is related to the Hausdorff dimension of the limit set of \(\Gamma\) by the formula \[ \lambda_0 = \alpha(1-\alpha). \] Hausdorff dimension of limit sets of Kleinian groups and eigenvalues of Laplacians on hyperbolic surfaces have been studied by a number of authors; compare R. S. Phillips and P. Sarnak [Acta Math. 155, 173–241 (1985; Zbl 0611.30037)], Curtis T. McMullen [Am. J. Math. 120, No. 4, 691–721 (1998; Zbl 0953.30026)] and Oliver Jenkinson and Mark Pollicott [Am. J. Math. 124, No. 3, 495–545 (2002; Zbl 1002.37023)]. In particular, algorithms for computing and estimating these quantities are known.
The author presents another algorithm for rigorously estimating \(\alpha\) (and hence \(\lambda_0\)) for the groups \(\Gamma_{\theta}\) defined above, and presents the resulting estimates for a range of parameters. He also compares his estimates with known asymptotics for \(\lambda_0\) as \(\theta\) tends to \(\pi/2\).

MSC:

30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
35P15 Estimates of eigenvalues in context of PDEs
37F30 Quasiconformal methods and Teichmüller theory, etc. (dynamical systems) (MSC2010)
37F35 Conformal densities and Hausdorff dimension for holomorphic dynamical systems
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