Ergodic Theory and Dynamical Systems



The Poincaré series of $\mathbb C\setminus\mathbb Z$


JON AARONSON a1 and MANFRED DENKER a2
a1 School of Mathematical Sciences, Tel Aviv University, 69978 Tel Aviv, Israel (e-mail: aaro@math.tau.ac.il)
a2 Institut für Mathematische Stochastik, Universität Göttingen, Lotzestr. 13, 37083 Göttingen, Germany (e-mail: denker@math.uni-goettingen.de)

Abstract

We show that the Poincaré series of the Fuchsian group of deck transformations of ${\mathbb C}\setminus{\mathbb Z}$ diverges logarithmically. This is because ${\mathbb C}\setminus{\mathbb Z}$ is a ${\mathbb Z}$-cover of the three horned sphere, whence its geodesic flow has a good section which behaves like a random walk on ${\mathbb R}$ with Cauchy distributed jump distribution and has logarithmic asymptotic type.

(Received December 2 1996)
(Revised February 25 1998)