Preface In the 1960’s, Steven Smale began an ambitious program to study the dynamics of smooth maps on manifolds [33]. This program has since become a substantial part of the theory of dynamical systems and can be found in many basic texts. In addition, many of the fundamental ideas and principles have had a large influence throughout dynamical systems. For example, see [7,16,20,30]. Smale introduced the notion of an Axiom A system: the main condition is that the map, when restricted to its set of non-wandering points, has a hyperbolic structure. Smale showed that the non-wandering of an Axiom A system can be canonically decomposed into finitely many disjoints sets, each of which is irreducible in a certain sense. Such sets are called basic sets. The study of the system then breaks down into the study of the individual basic sets and the problem of how the rest of the manifold is assembled around them. An important subtlety that Smale realized from the start was that the basic sets were not typically submanifolds, but rather some sort of fractal object. Moreover, the hyperbolic nature of the dynamics on the basic set (along with the assumption that the periodic points are dense in the non-wandering set) created the conditions which are now usually referred to as chaos. This study forms the mathematical foundations for chaos, which has had a profound impact in many areas of science and beyond. David Ruelle introduced the notion of a Smale space in an attempt to axiom- atize the dynamics of an Axiom A system, when restricted to a basic set (or the non-wandering set) [31]. This involved giving a definition of hyperbolicity for a homeomorphism of a compact metric space. From our point of view here, there are two essential differences between the non-wandering set for an Axiom A system and a Smale space. The first is that in the former, every point is non-wandering, which need not be the case for a Smale space. Indeed our constructions will involve a number of Smale spaces with wandering points. Secondly, a Smale space can be described without seeing it as a subset of a manifold (to which the dynamics extends in an appropriate manner). Very early on, a particular class of Smale spaces took a prominent role: the shifts of finite type. As the name suggests, these are dynamical systems of a highly combinatorial nature and the spaces involved are always totally disconnected. They had been studied previously, but in this context they appeared in two essential ways. First, Smale showed how such systems could appear as basic sets for Axiom A system the full 2-shift is a crucial feature in Smale’s horseshoe. Secondly, they could be used to code more complicated systems. This idea is originally credited to Hadamard for modelling geodesic flows [19] and was pursued in work of Morse [26], Morse and Hedlund [27] and many others. For a modern treatment, see [4]. For Axiom A systems, Bowen [6], building on work of Adler-Weiss, Sinai and others, proved that every basic set (or every Smale space) is the image of a shift of finite vii
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