Undecidability in Rn: Riddled Basins, the KAM Tori, and the Stability of the Solar System

Abstract

Some have suggested that certain classical physical systems have undecidable long-term behavior, without specifying an appropriate notion of decidability over the reals. We introduce such a notion, decidability in μ (or d-μ) for any measure μ, which is particularly appropriate for physics and in some ways more intuitive than Ko's (1991) recursive approximability (r.a.). For Lebesgue measure λ, d-λ implies r.a. Sets with positive λ-measure that are sufficiently "riddled" with holes are never d-λ but are often r.a. This explicates Sommerer and Ott's (1996) claim of uncomputable behavior in a system with riddled basins of attraction. Furthermore, it clarifies speculations that the stability of the solar system (and similar systems) may be undecidable, for the invariant tori established by KAM theory form sets that are not d-λ.