# American Institute of Mathematical Sciences

January  1997, 3(1): 107-116. doi: 10.3934/dcds.1997.3.107

## A billiard in the hyperbolic plane with decay of correlation of type $n^{-2}$

 1 Department of Mathematics, Campus de Beaulieu, Université de Rennes I, 35042 Rennes - Cedex, France 2 Instituto de Matemática, Universidade Federal RGS Av. Bento Gonçalves 9500, 91509 Porto Alegre - RS, Brazil

Received  August 1996 Published  October 1996

We consider billiard trajectories on the geodesic triangle $D$ in the hyperbolic half-plane with internal angles $0,0$ and $\pi /2$ at the vertices $\infty,1$, and $i$. The billiard map of $D$ sends a given trajectory $\gamma$ with the boundary $\delta D$ of $D$ to the next intersection point of $\gamma$ with $\delta D$.
By choosing an appropriate cross section for the billiard map, we show that the decay of correlation of the first return map is slower than $n^-2$. As a by-product, we enumerate the billiard trajectories in terms of their cutting sequences and relate the boundary expansion of the billiard map to continued fractions with even partial quotients.
Citation: M. Bauer, A. Lopes. A billiard in the hyperbolic plane with decay of correlation of type $n^{-2}$. Discrete and Continuous Dynamical Systems, 1997, 3 (1) : 107-116. doi: 10.3934/dcds.1997.3.107
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