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 billiards.
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