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Abstract. A real is called recursively enumerable if it is the limit of a recursive, increasing, converging sequence of rationals. Following Solovay 23 and Chaitin 10 we s a y that an r.e. real dominates an r.e. real if from a good approximation of from below one can compute a good approximation of from below. We shall study this relation and characterize it in terms of relations between r.e. sets. Solovay's 23 -like numbers are the maximal r.e. real numbers with respect to this order. They are random r.e. real numbers. The halting probability o f a universal self-delimiting Turing machine Chaitin's number, 9 is also a random r.e. real. Solovay showed that any Chaitin numberis -like. In this paper we show that the converse implication is true as well: any -like real in the unit interval is the halting probability of a universal self-delimiting Turing machine.

The main results of the paper are two eective v ersions of the Riemann mapping theorem. The rst, uniform version is based on the constructive proof of the Riemann mapping theorem by Bishop and Bridges and formulated in the computability framework developed by Kreitz and Weihrauch. It states which topological information precisely one needs about a nonempty, proper, open, connected, and simply connected subset of the complex plane in order to compute a description of a holomorphic bijection from this set onto the unit disk, and vice versa, which topological information about the set can be obtained from a description of a holomorphic bijection. The second version, which is derived from the rst by considering the sets and the functions with computable descriptions, characterizes the subsets of the complex plane for which there exists a computable holomorphic bijection onto the unit disk. This solves a problem posed by Pour{El and Richards. We also show that this class of sets is strictly larger than a class of sets considered by Zhou, which solves an open problem posed by him. In preparation, recursively enumerable open subsets and closed subsets of Euclidean spaces are considered and several eective results in complex analysis are proved.

The topological complexity of algorithms is studied in a general context in the first part and for zero-finding in the second part. In the first part the level of discontinuity of a function f is introduced and it is proved that it is a lower bound for the total number of comparisons plus 1 in any algorithm computing f that uses only continuous operations and comparisons. This lower bound is proved to be sharp if arbitrary continuous operations are allowed. Then there exists even a balanced optimal computation tree for f. In the second part we use these results in order to determine the topological complexity of zero-finding for continuous functions f on the unit interval with f (0) и f (1) Ͻ 0. It is proved that roughly log 2 log 2 Ϫ1 comparisons are optimal during a computation in order to approximate a zero up to . This is true regardless of whether one allows arbitrary continuous operations or just function evaluations, the arithmetic operations ͕ϩ, Ϫ, *, /͖, and the absolute value. It is true also for the subclass of nondecreasing functions. But for the subclass of increasing functions the topological complexity drops to zero even for the smaller class of operations.

A real is called recursively enumerable if it is the limit of a recursive, increasing, converging sequence of rationals. Following Solovay 23 and Chaitin 10 we s a y that an r.e. real dominates an r.e. real if from a good approximation of from below one can compute a good approximation of from below. We shall study this relation and characterize it in terms of relations between r.e. sets. Solovay's 23like numbers are the maximal r.e. real numbers with respect to this order. They are random r.e. real numbers. The halting probability o f a universal self-delimiting Turing machine Chaitin's number, 9 is also a random r.e. real. Solovay showed that any Chaitin numberis -like. In this paper we show that the converse implication is true as well: any -like real in the unit interval is the halting probability of a universal self-delimiting Turing machine.

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