Construction of strictly ergodic systems

Abstract

Recent results of Jewett [2] and Krieger [3] assert, that every ergodic measure theoretic dynamical system on a Lebesgue space is isomorphic to a strictly ergodic dynamical system which is embedded in a shift space with an alphabet, the minimum length of which is determined by the entropy of the transformation. However, the corresponding strictly ergodic system is not given in a constructive way. Therefore there remains the problem of constructing strictly ergodic systems. It turns out that shift spaces with finite alphabet are particularly well suited for such constructions because of their simple topological properties. In this way, Hahn and Katznelson ([1]) have found strictly ergodic systems with arbitrarily large entropies in shift spaces. In this paper, we obtain the same result by a new and considerably simpler construction. Moreover, for constructively given h <log k, we construct a system with entropy h in a shift space with alphabet of length k. Also, in a shift space with compact infinite alphabet, we construct a system with infinite entropy. For the constructions we use only permutations of block systems. A considerable simplification of the proofs is possible by the result of Parry [5], that in our case topological and measure theoretic entropy are the same. In a further paper we shall give a construction for strictly ergodic K-systems.