Symbolic Dynamics and Transformations of the Unit Interval

0. Introduction. This paper extends some results from [1] and applies them to certain transformations of the unit interval. §§2-4 are concerned with symbolic dynamics and §§5-6 are concerned with their application to the proof of our main theorem which states sufficient conditions for a piecewise continuous transformation of the unit interval to be conjugate to a (uniformly) piecewise linear transformation. The main result includes a classical theorem of Poincaré-Denjoy [2] on homeomorphisms of the circle onto itself. It also provides a partial answer to a question of Ulam's [3] concerning the possibility of piecewise linearising continuous transformations of the unit interval. This problem was also mentioned by Stein and Ulam in [4], together with the remark that necessary conditions can be given in terms of the trees of points, but that no meaningful sufficient conditions are known. In the same work a few special examples are examined. Our main theorem also has a bearing on certain transformations discussed by Rényi [5]. In §§2-4 we consider the shift transformation acting on a compact invariant subset of the space of one-way infinite sequences of symbols chosen from a finite set. The shift transformation on such a set is continuous but not necessarily open. If X, T are the compact invariant set and the shift transformation, respectively, we refer to (X, T) as a symbolic dynamical system [6]. For a symbolic dynamical system (X, T) we define a number called the absolute entropyO) which dominates the entropy of T with respect to each normalised T invariant Borel measure, and show that if T is regionally transitive then there is always one invariant measure with respect to which the entropy of T equals the absolute entropy of T. When T is open, (or equivalently, when (X, T) is an intrinsic Markov chain) this "maximal" measure is unique. A further theorem states that, under certain conditions, there exists a normalised Borel measure with respect to which T acts in a "linear" fashion. In §§5-6 we apply this latter theorem to certain transformations of the unit interval and obtain our main result, Theorem 5.