  Since the seminal work of Sinai one studies chaotic properties of planar
billiards tables. Among them is the study of decay of correlations for these
tables. There are examples in the literature of tables with exponential and
even polynomial decay. However, until now nothing is known about mixing
properties for billiard tables with non-compact cusps. There is no consensual
definition of mixing for systems with infinite invariant measure. In this paper
we study geometric and ergodic properties of billiard tables with a non-compact
cusp. The goal of this text is, using the definition of mixing proposed by
Krengel and Sucheston for systems with invariant infinite measure, to show that
the billiard whose table is constituted by the x-axis and and the portion in
the plane below the graph of $f(x)=\frac{1}{x+1}$ is mixing and the speed of
mixing is polynomial.
