From the Publisher:
Although it originated as a method to study general dynamical systems, symbolic dynamics is useful in coding for data storage and transmission as well as in linear algebra. Requiring only a undergraduate knowledge of linear algebra, this first general textbook includes over 500 exercises.

An Introduction to Symbolic Dynamics and Coding

Gibbs Measures.- General Thermodynamic Formalism.- Axiom a Diffeomorphisms.- Ergodic Theory of Axiom a Diffeomorphisms.

https://www.cpht.polytechnique.fr/sites/default/files/Bowen_LN_Math_470_second_ed_v2013.pdf

Equilibrium states and the ergodic theory of Anosov diffeomorphisms

Topological entropy há(T) is defined for a uniformly continuous map on a metric space. General statements are proved about this entropy, and it is calculated for affine maps of Lie groups and certain homogeneous spaces. We compare hd(T) with measure theoretic entropy h(T); in particular h(T) = hd(T) for Haar measure and affine maps Ton compact metrizable groups. A particular case of this yields the wellknown formula for h(T) when T is a toral automorphism. Introduction. We shall study topological entropy, concentrating on its relation to measure theoretic entropy and algebraic examples. Our topological entropy hd(T) is defined (in §2) for a uniformly continuous map F on a metric space (X, d). In [1] a topological entropy h(T) was defined for a continuous map on a compact topological space; if the space is compact metric then h(T)—hd(T). An essential part of this paper is the computation of hd(T) for certain maps on noncompact spaces. Suppose p is a Borel measure on p(X) = l, and p is F-invariant (i.e. p(T~x(A)) =p(A) for every Borel set A). One can then define a measure theoretic entropy hu(T) as follows: Call a={Au ..., Ar} a (finite) measurable partition of A\" if the A¡ are disjoint measurable subsets of X covering X. Now set Hja) = 2 -4mc\\ T-kAik) iogp(mn t-*a\\ Then the limit hß(T, a) = limm_00 (\\/m)HJa) exists and one defines hu(T) = sup {hu(T, a) : a is a finite measurable partition of X}. (See [6] for details about measure theoretic entropy.) Two points in X are separated by a = {A±,..., Ar} provided they lie in different ^i's. We shall use the following fact to compute entropy : Fact (see [6]). Let {ak}k = 0 be a sequence of measurable partitions of X satisfying the following property: If x, ye X are distinct there is an n(x, y) such that ak separates x and y whenever k S: n(x, y). Then hu(f) = supfc h(T, ak). As is generally known, if T: G -> G is a surjective endomorphism of a compact metrizable group, then F preserves Haar measure p. For such a F we show that the Received by the editors October 17, 1969 and, in revised form, February 2, 1970. AMS 1969 subject classifications. Primary 2870, 2875; Secondary 5482.

/pdf/entropy-for-group-endomorphisms-and-homogeneous-spaces-58urtdfuju.pdf

Entropy for group endomorphisms and homogeneous spaces

Let X(Se) be the set of all bisequences over a symbol set 6 a, where 1 < card S# < 0% and let cr be the shift transformation. If the product topology induced by the discrete topology of 6: is assigned to X(6a), X(6 a) is homeomorphic to the Cantor discontinuum and ~ is a homeomorphism of X(6Q onto X(6Q. The discrete flow (X(SQ, a) is the symbolic flow over 5: or the shift dynamical system over S a. The shift dynamical system (X(S:), a) has been analyzed rather thoroughly, both in its topological and in its measure-theoretic aspects [1, 7, 9, 10, 13, 14]. Recent work of Smale [27] shows that the shift dynamical system is ubiquitous. Any closed subset of X(SQ which is invariant under a defines a subdynamical system. There is an endless variety of these and they have served as useful models to indicate possible structures of dynamical systems, particularly minimal sets [7, 8, 9, 10, 11, 12, 15, 20, 22, 23, 24, 25]. These systems are characterized by the fact that the phase space is totally disconnected and the transformation is expansive (Section 2). However, there are dynamical systems, notably geodesic flows on compact manifolds of negative curvature, for which the phase space is a manifold, yet the orbits can be characterized by symbolic bisequences [2, 3, 4, 5, 20, 21, 28]. Properties of such dynamical systems and their subdynamical systems can be determined from knowledge of the properties of symbolic flows. A natural question in connection with any dynamical system is that of the existence and properties of continuous transformations which commute with the group action. In the case of the system (X(S:), ~), an obvious example of such a transformation is obtained by simply permuting the symbols. A generalization of this is to define a mapping of blocks (words) of symbols of a specified length into single symbols and to extend this mapping in a natural manner to infinite sequences. It has been shown by Curtis, Hedlund and Lyndon that these mappings, composed with powers of the shift, constitute the entire class of continuous transformations which commute with the shift (Section 3). This fundamental

Endomorphisms and automorphisms of the shift dynamical system

In this paper we introduce the concept of a Gibbs measure, which generalizes the concept of an equilibrium Gibbs distribution in statistical physics. The new concept is important in the study of Anosov dynamical systems. By means of this concept we construct a wide class of invariant measures for dynamical systems of this kind and investigate the problem of the existence of an invariant measure consistent with a smooth structure on the manifold; we also study the behaviour under small random excitations as . The cases of discrete time and continuous time are treated separately.

https://iopscience.iop.org/article/10.1070/RM1972v027n04ABEH001383/pdf

Gibbs measures in ergodic theory

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Zeta functions and the periodic orbit structure of hyperbolic dynamics

/pdf/intrinsic-markov-chains-4eajvwuaan.pdf

Intrinsic Markov chains

Entropy and generators in ergodic theory

Preface Introduction 1. The principal ergodic theorems 2. Martingales and the ergodic theorem of information theory 3. Mixing 4. Entropy 5. Some examples Appendix References Further literature Index.

Topics in Ergodic Theory

transformation. The main result includes a classical theorem of Poincare-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 Renyi [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 entropy(') 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.

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William Parry

Papers

Zeta functions and the periodic orbit structure of hyperbolic dynamics

Intrinsic Markov chains

Entropy and generators in ergodic theory

Topics in Ergodic Theory

Symbolic dynamics and transformations of the unit interval