  Let $f: [0, +\infty) \to (0, +\infty)$ be a sufficiently smooth convex
function, vanishing at infinity. Consider the planar domain $Q$ delimited by
the positive $x$-semiaxis, the positive $y$-semiaxis, and the graph of $f$.
  Under certain conditions on $f$, we prove that the billiard flow in $Q$ has a
hyperbolic structure and, for some examples, that it is also ergodic. This is
done using the cross section corresponding to collisions with the dispersing
part of the boundary. The relevant invariant measure for this Poincar\'e
section is infinite, whence the need to surpass the existing results, designed
for finite-measure dynamical systems.
