Ethan M. Coven

Endomorphisms of irreducible subshifts of finite type

Introduction. The properties of endomorphisms of the full shift dynamical system are described by Hedlund in [9]. His proofs are based on the very nice combinatorial properties of the full shift. However, the combinatorial structure of a symbolic flow is, in general, not susceptible to the kind of analysis done in [9]. For this reason, there are relatively few results about endomorphisms of symbolic flows other than full shifts. In this paper, we investigate the properties of endomorphisms of a class of symbolic flows known as irreducible subshifts of finite type. This class, which contains all full shifts, is in some sense a more appropriate class to study than the class of full shifts. Irreducible subshifts of finite type occur naturally in the work of the Smale school on Axiom A diffeomorphisms (see [3], [4], [12]). They were introduced by Parry [I 1], who called them intrinsic Markov chains. They are examples of intrinsically ergodic flows, i.e., flows having a unique invariant measure such that the topological entropy of the flow is finite and equal to the measure-theoretic entropy with respect to the distinguished measure. For an irreducible subshift of finite type, the value of this measure on a basic cylinder set is easily computed. We first establish some properties of intrinsically ergodic symbolic flows and their endomorphisms. Then we prove the equivalence of certain properties of an endomorphism of an irreducible subshift of finite type (e.g., being onto, being finite-to-one, preserving the distinguished measure). Finally, we show how to extend Hedlunds results on inverses of onto endomorphisms to endomorphisms of irreducible subshifts of finite type. The authors wish to thank G. A. Hedlund for many hours of stimulating discussion.

Non-wandering sets of the powers of maps of the interval

We show that, for maps of the interval, the non-wandering set of the map coincides with the non-wandering set of each of its odd powers, while the nonwandering set of any of its even powers can be strictly smaller.

The structure of substitution minimal sets

Substitutions of constant length on two symbols and their corresponding minimal dynamical systems are divided into three classes: finite, discrete and continuous. Finite substitutions give rise to uninteresting systems. Discrete substitutions yield strictly ergodic systems with discrete spectra, whose topological structure is determined precisely. Continuous substitutions yield strictly ergodic systems with partly continuous and partly discrete spectra, whose topological structure is studied by means of an associated discrete substitution. Topological and measure-theoretic isomorphisms are studied for discrete and continuous substitutions, and a complete topological invariant, the normal form of a substitution, is given. 0. Introduction. Let 0 denote a substitution of constant length n on the symbols 0 and 1. Our main objective is the classification of the dynamical systems (0,, r) arising from such substitutions. A substitution 0 is either finite, discrete, or continuous as defined in ?3. Finite substitutions give rise to dynamical systems all of whose minimal sets are periodic orbits. Discrete and continuous substitutions both give rise to strictly ergodic dynamical systems with discrete and partially continuous spectra respectively. More specifically, if 0 is discrete, then V. is the orbit-closure of a Toeplitz bisequence, and if 0 is continuous, then (C is the orbit-closure of an extended generalized Morse sequence. Dynamical systems belonging to discrete substitutions are measure-theoretically isomorphic if and only if the lengths of the substitutions have the same prime factors. If 0 is discrete, we are able to give an explicit construction of (0,, r) as an almost one-to-one extension of the n-adic system (Z(n), r). We associate with 0 an object called a group system which is a complete invariant for topological isomorphism of (0,, r). The concept of normal form for a discrete substitution is defined and it is shown that discrete substitutions of the same length possess topologically isomorphic dynamical systems if and only if they have the same normal form. To each continuous substitution 0 we associate a discrete substitution 0 of the same length such that (0, r) is a distal extension of (06, r). Each fibre consists of two points which are mirror images of each other. A normal form for continuous Received by the editors May 22, 1970. AMS 1970 subject classifications. Primary 28A65, 54H20.

Transitivity and the centre for maps of the circle

We study the dynamics of continuous maps of the circle with periodic points. We show that the centre is the closure of the periodic points and that the depth of the centre is at most two. We also characterize the property that every power is transitive in terms of transitivity of a single power and some periodic data.

ω-limit sets for maps of the interval

Let f denote a continuous map of a compact interval to itself, P(f) the set of periodic points of f and Λ(f) the set of ω-limit points of f. Sarkovskǐi has shown that Λ(f) is closed, and hence ⊆Λ(f), and Nitecki has shown that if f is piecewise monotone, then Λ(f)=. We prove that if x∈Λ(f)−, then the set of ω-limit points of x is an infinite minimal set. This result provides the inspiration for the construction of a map f for which Λ(f)≠.

Homoclinic and non-wandering points for maps of the circle

For continuous maps ƒ of the circle to itself, we show: (A) the set of nonwandering points of ƒ coincides with that of ƒ n for every odd n ; (B) ƒ has a horseshoe if and only if it has a non-wandering homoclinic point; (C) if the set of periodic points is closed and non-empty, then every non-wandering point is periodic.

The Symbolic Dynamics Of Multidimensional Tiling Systems

We prove a multidimensional version of the theorem that every shift of finite type has a power that can be realized as the same power of a tiling system. We also show that the set of entropies of tiling systems equals the set of entropies of shifts of finite type.

Periods of some nonlinear shift registers

Abstract We determine the set of all possible least periods of shift register sequences for non-linear feedback functions of the form f ( x 0 ,…, x m −1 ) = x 0 + Π i =1 k ( x i + b i ) where m ⩾ k + 1 ⩾ 3 and the least period of the k -block b 1 … b k itself.

A characterization of the Morse minimal set up to topological conjugacy

We establish necessary and sufficient conditions for a dynamical system to be topologically conjugate to the Morse minimal set, the shift orbit closure of the Morse sequence, and conditions for topological conjugacy to the closely related Teoplitz minimal set.

Every compact metric space that supports a positively expansive homeomorphism is finite

We give a simple proof of the title. A continuous map f : X → X of a compact metric space, with metric d, is called positively expansive if and only if there exists ǫ > 0 such that if x 6= y, then d(f(x), f(y)) > ǫ for some k ≥ 0. Here exponentiation denotes repeated composition, f = f ◦ f , etc. Any such ǫ > 0 is called an expansive constant. The set of expansive constants depends on the metric, but the existence of an expansive constant does not. Examples include all one-sided shifts and all expanding endomorphisms, e.g., the maps z 7→ z, n 6= 0,±1, of the unit circle. None of these maps is one-to-one, and with good reason. It has been known for more than fifty years that every compact metric space that supports a positively expansive homeomorphism is finite. This was first proved by S. Schwartzman [8] in his 1952 Yale dissertation. The proof appears in [4, Theorem 10.30]. (A mistake in the first edition was corrected in the second edition.) Over the years it has been reproved with increasingly simpler proofs. See [5],[6],[7]. The purpose of this paper is to give another, even simpler, proof of this result. The idea behind our argument is not new. After discovering the proof given in this paper, the authors learned from W. Geller [3] that the basic idea of the proof had been discovered in the late 1980s by M. Boyle, W. Geller, and J. Propp. Their proof appears in [6]. The result also follows from the theorem at the end of Section 3 of [1]. The authors of that paper may have been unaware of this consequence of their theorem, which they called “a curiosity based on the techniques of this work.” That the result follows from the theorem in [1] was recognized by B. F. Bryant and P. Walters [2]. Nonetheless, the fact that this result has a simple proof remains less well-known than it should be, and so publishing it in this Festschrift is appropriate. Theorem. Every compact metric space that supports a continuous, one-to-one, positively expansive map is finite. Remark. Our statement does not assume that the map is onto. Such maps are sometimes called “homeomorphisms into.” We have written the proof so that the fact that f is one-to-one is used only once. Department of Mathematics, Wesleyan University, Middletown, CT 06459, USA, e-mail: ecoven@wesleyan.edu ; mkeane@wesleyan.edu AMS 2000 subject classifications: primary 37B05; secondary 37B25.