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Let G be a discrete subgroup of SL(2, C)/{• 1}. Then G operates as a discontinuous group of isometrics on hyperbolic 3-space, which we regard as the open unit ball B a in Euclidean 3-space E a. G operates on S 2, the boundary of B a, as a group of conformal homeomorphisms, but it need not be discontinuous there. The set of points of S 2 at which G does not act discontinuously is the limit set A(G).If we fix a point 0 in B a, then the orbit of 0 under G accumulates precisely at A(G).The approximation is, however, not uniform. We distinguish a class of limit points, called points o/aproximation, which are approximated very well by translates of 0. The set of points of approximation includes all loxodromic (including hyperbolic) fixed points, and includes no parabolic fixed points. In w 1 we give several equivalent definitions of point of approximation, and derive some properties. We remark that these points were first discussed by Hedlund [7].Starting with a suitable point 0 in B a, we can construct the Dirichlet fundamental polyhedron P0 for G. It was shown by Greenberg [5] that even if G is finitely generated, P0 need not have finitely many sides. Our next main result, given in w 2, is that if P0 is finite-sided, then every point of A(G) is either a point of approximation or a cusped parabolic fixed point (roughly speaking a parabolic fixed point is cusped if it represents the right number of punctures in (S2-A(G))/G).The above theorem has several applications: one of these is a new proof of the following theorem of Ahlfors [1].I/Po has finitely many sides, then the Euclidean measure o/ A(G) is either 0 or 4~.Our next main result, given in w 3, is that the above necessary condition for P0 to have finitely many sides is also sufficient. In fact, we prove that any convex fundamental polyhedron G has finitely many sides if and only if A(G) consists entirely of points of 1-742908 Acta mathematica 132. Imprim6 1r 18 Mars 1974

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We show that, providing a metric space $X$ has a boundary that is in some sense similar to the boundary of hyperbolic space, the iterates of a contraction $f:X\to X$ converge locally uniformly to a point in, or on the boundary of, $X$. This generalises the Denjoy–Wolff theorem for analytic self-maps of the unit disc in the complex plane, and also shows that if $D$ is a bounded strictly convex subdomain of ${\Bbb R}^n$, then any contraction of $D$ with respect to the Hilbert metric of $D$ converges to a point in the closure of $D$.

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