Symbolic dynamics and transformations of the unit interval

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.