Extension to the Boundary
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In this chapter we study the boundary extension of the iterates of a semigroup and of the associated Koenigs function. After studying the impression and the principal part of prime ends of domains defined by Koenigs functions, we prove that every Koenigs function and every iterate of a semigroup have non-tangential limit at every boundary point. Moreover, the semigroup functional equation and the functional equation defined by the canonical model extend in the non-tangential limits sense up to the boundary. In the last part of the chapter we analyze conditions for a tout court continuous extension of iterates of a semigroup up to the boundary.