Computable Chaotic Orbits of Ergodic Dynamical Systems

Abstract

The central theme of this Institute is scaling in disordered systems, especially the use of both regular (periodic) and ‘random’ fractals to describe the scaling properties. In a related set of lectures, Wolfram¹ has shown how both regular and pseudo-random states can be generated systematically via simple integer maps called cellular automata. The question whether a given problem is ‘decidable or intractable’^1-b was touched upon in a general way. For example, the question whether a given cellular automaton generates a truly random sequence of bits when no pattern can be found may be undecideable: it is always possible that an algorithm generates a pattern that is sufficiently complex that we may fail to decode it, yet it may pass all standard tests for randomness. We regard this as an example of the sort of limitation upon computability that is suggested by the work of Turing.^2,2-b