The gap distribution of directions in some Schottky groups

Xin Zhang

doi:10.3934/jmd.2017019

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2017, Volume 11: 477-499. Doi: 10.3934/jmd.2017019

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The gap distribution of directions in some Schottky groups

Xin Zhang

Department of Mathematics, University of Illinois, 1409 W, Green Street, Urbana, IL 61801, USA

Received: April 21, 2016

Revised: May 23, 2017

Published: October 2017

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Abstract

We prove the existence and some properties of the limiting gap distribution for the directions of some Schottky group orbits in the Poincaré disk. A key feature is that the fundamental domains for these groups have infinite area.

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Mathematics Subject Classification: Primary: 11J71; Secondary: 11N45, 37F35, 22E40.

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Figure 1. A fundamental domain of a group of hyperbolic reflections

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Figure 2. The family of $C_{\gamma}$ 's (hued circles)

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Figure 3. The plot for the gap distribution function $F_{10^4,\partial{\mathbb{D}}}$ , for the example illustrated in Figure 2

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Figure 4. The histograms of $F_{T,\mathscr{I}}'$ of different $T$ , for the example illustrated in Figure 2

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Figure 5. Reflecting a circle

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Figure 6. Case (a)

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Figure 7. Case (b)

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