CHAPTER 1 Summary The aim of this paper is to define a type of homology theory for Smale spaces, which include the basic sets for Smale’s Axiom A systems. The existence of such a theory was conjectured by Rufus Bowen [7]. Our approach is based on Krieger’s dimension group invariant for shifts of finite type. We will use this chapter as an introduction to the concepts and a summary of the paper, stating the main new definitions and results. We will be concerned with Smale spaces, as defined by David Ruelle [31] we summarize here informally, the precise definition will be given in Definition 2.1.6. We have a compact metric space (X, d) and a homeomorphism, ϕ, of X. Such a topological dynamical system is called a Smale space if it possesses local coordinates of contracting and expanding directions. Roughly, to any point x in X and sufficiently small, there exist subsets Xs(x, ) and Xu(x, ) called the local stable and unstable sets at x. Their Cartesian product is homeomorphic to a neighbourhood of x. The parameter controls the diameter of these sets and, as it varies, these product neighbourhoods form a neighbourhood base at x. The contracting/expanding nature of ϕ is described by the condition that there is a constant 0 λ 1 such that d(ϕ(y),ϕ(z)) λd(y, z), for all y, z Xs(x, ), d(ϕ−1(y),ϕ−1(z)) λd(y, z), for all y, z Xu(x, ). There is also a condition roughly indicating that the local product structure is invariant under ϕ. Ruelle’s precise definition involves the existence of a map [x, y] defined on pairs (x, y) which are sufficiently close and with range in X. The idea is that [x, y] is the unique point in the intersection of the local stable set of x and the local unstable set of y. Ruelle provided appropriate axioms for this map. Given the map [·, ·], the set Xs(x, ) consists of those y with d(x, y) and [x, y] = y, or equivalently (via the axioms) [y, x] = x. Ruelle’s main objective in giving such a definition was to provide an axiomatic framework for studying the basic sets for Smale’s Axiom A systems [7,33]. While Smale’s original definition is in terms of smooth maps of Riemannian manifolds, the basic sets typically have no such smooth structure. In addition, there are examples of Smale spaces, some of which will play an important part in this paper, which do not appear in any obvious way as a basic set sitting in a manifold. Moreover, basic sets, by their definition, are irreducible while Smale spaces need not be. Again, there will be many Smale spaces which play an important part here which fail to be non-wandering. In a Smale space, two points x and y are stably equivalent if lim n→+∞ d(ϕn(x),ϕn(y)) = 0. 1
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