Effective Real Dynamics

Abstract

The study of computability in analysis has a long history, going back to the papers of Lacombe [6] in the 1950’s. There has been much work on the connection between recursive function theory and computable analysis. One key result which we will use here is a theorem of Nerode’s from [7] that the existence of a recursive continuous function mapping a real a to a real b implies that b is truth-table reducible to a. Another connection which we will use is given in the papers of Soare [11, 12] on recursion theory and Dedekind cuts, where the effectively closed real intervals are characterized. One important aspect of computable analysis is the search for effective versions of classical theorems. This is in the spirit of the well-known Nerode program for applying recursion theory to mathematics. As an example, if K is a closed subset of the real line ℜ, then the distance function δ_K’ defined by δ_K(x) = min{|x − y|: y ∈ K} is continuous. Thus the question arises whether an effectively closed set K has an effectively continuous (i.e., recursive) distance function. (In general, the answer is no.) Closed sets play an important rôle in the study of analysis. For example, the set of zeros of a continuous function F is a closed set, as is the set of fixed points of F. We are particularly interested in the rôle of effectively closed, or Π⁰ ₁ classes. Π₁ ⁰ classes are important in the applications of recursion theory and have been studied extensively. (See [4] for a survey of results.)