Diagonalization, uniformity, and fixed-point theorems

We derive new fixed-point theorems for subrecursive classes, together with a theorem on the uniformity of certain reductions, from a general formulation of the technique of delayed diagonalization. This formulation extends the main theorem of U. Schöning (Theoret. Comput. Sci. 18 (1982), 95-103) to cases which involve infinitely many diagonal classes C_k, and which allow each C_k to contain uncountably many members. The main technical work ties the familiar concept of a witness function directly to the often-studied Cantor-set topology on languages, and provides a "delay construction" which refines those due to Schöning, S. Breidtbart, and D. Schmidt. Our "a.e." fixed-point theorems do not require that the "programming system" for the subrecursive class in question be well-behaved; we compare them to results which do. The other theorem is similar to the "uniform boundedness theorem" of classical analysis, and extends work of J. Grollmann and A. Selman.