Complexity and Randomness of Recursive Discretizations of Dynamical Systems

We begin by giving a heuristie description of the approach taken in ibis. paper. Suppose (X,T) is a dynamical system with a distinguished subset of a11 "computable" points in X. With the definition of "computability" taken in ibis article ibis will be automatically aT-invariant set. Any of its subsets (finite 01' infinite) is ca11ed a recursive discl'etization of (X, T). What information about the dynamics of (X, T) can Olle re cover from a recursive discretization? If we try to simulate (X, T) on a computer then we consider finite subsets of computable points and orbits of these points under a computer approxirnation of the mapping T. What information about the dynamics of (X, T) ean one reeover from a computer simulation? In both Gases the information we are interested in recovering is the ergodic behavior of (X, T), i.e. suitable time averages, Lyapunov exponents,etc. Different recursive discretizations can give rise to quite different behavior. Can one fihd natural discretizations whieh lead to the expected (classical) behavior? Dur motivation is partly practical-do computer simulations work?partly philosophical: what kind of ergodic theory can be düne in a constructive manner? Both discretization and simulation are the object of many mathematical and physical articIes. The study of discretization by periodic orbits is an old erle. For example, in ergodic theory it is we11 known thai Gibbs measures can be described by the set of periodic points [B02]. Periodic orbits are regarded as a linkage between quantum and classical systems inthe search für quantum chaos. The interpretation of statistical properties restricted to the periodic points hag led to confusion in the literature, due to the fact thai in the classical sense ibis motion cannot be understood as chaotic