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In the previous chapters we studied boundary points which are either fixed or the initial points of maximal invariant curves for a semigroup. In this chapter we examine the other points, which turn out to be contact points, and we show that super-repelling fixed points can be divided into two separated sets: those which are the landing point of a backward orbit and those which are the initial point of a maximal contact arc (in the latter case they are also critical points for the infinitesimal generators). We also discuss the behavior of the Koenigs function and the infinitesimal generator at the end points of maximal contact arcs. The chapter ends with some examples and, in particular, with the construction of a semigroup with an uncountable set of super-repelling fixed points.