Section: New Results
Computations
Participant : Mathieu Hoyrup.
Inversion of computable functions
We strengthen the preceding result by making and computable. This result is a particular case of a more general problem. In many situations an operator can be computed but can hardly be reversed: given , cannot always we recovered (computed) even when is one-to-one. We introduce a strong notion of discontinuity for the inverse of and prove that it entails the existence of a non-computable such that is computable. Our result on the ergodic decomposition can be derived by applying our general result to the operator which is computable but difficult to reverse. At the same time we prove a significant improvement of a classical result of Pour-El and Richards [67] about the computability of linear operators. The paper [26] is currently submitted.
Computability and measure theory.
We study the constructive content of the Radon-Nikodym theorem, show that it is not computable in general and precisely locate its non-computability in the Weihrauch lattice. The paper [15] appeared in the first issue of the new journal Computability.