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In this paper we prove that on a simple surface where the metric is $C^{17}$,the scattering relation determines the Dirichlet to Neumann map (DN map) - aknown result for the case when the metric is smooth. For metrics with finitedifferentiability we had to modified each technical result used in the originalproof; such as properties of the exit time function and the characterization of$C_\alpha$ space. Moreover, surjectivity of $I^*$ in the original proofrequired the use of microlocal analysis on the normal operator $I^*I$ - whichis not a standard pseudodifferential operator for metrics with finiteregularity. Finally, using the injectivity of $I$ on Lipschitz one forms forsimple $C^{1,1}$ manifolds we prove an equivalent characterization of harmonicconjugacy using operators defined by the scattering relation to prove thetitular result. We also prove that the boundary distance function determinesthe metric at the boundary (which in turns determines the scattering relation)for a closed disk even when the metric is only $C^{1,1}$ and the exponentialmap is only Lipschitz and does not preserve tangent vectors or differentialspointwise.

Determination of DN map from the scattering relation for simple surfaces at low regularity