  This work constructs symbolic dynamics for non-uniformly hyperbolic surface
maps with a set of discontinuities $D$. We allow the derivative of points
nearby $D$ to be unbounded, of the order of a negative power of the distance to
$D$. Under natural geometrical assumptions on the underlying space $M$, we code
a set of non-uniformly hyperbolic orbits that do not converge exponentially
fast to $D$. The results apply to non-uniformly hyperbolic planar billiards,
e.g. Bunimovich stadia.
