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Anosov diffeomorphisms of flat manifolds

Let M be a compact differentiable manifold without boundary. A Riemannian structure on II is called flat if all sectional curvatures vanish at each point; then M is called a flat manifold A diffeomorphism f :→M is called an Anosov diffeomorphism. if for some (and hence any) Riemannian metric on M there exist constants c > 0, ƛ< 1 such that at any point m of M the tangent space TMm decomposes as the direct sum of a contracting part and an expanding part; more precisely TMm = Es Φ Eu, where ║Tfrv ║≤,cƛr║v║ for all v E Es and all integers r > 0 and ║Tfrw║ ≤ c ƛr║w║ for all W E Eu and all integers r > 0 (the letters s and u stand as usual, for stable and unstable; they are also used for dimensions of the spaces involved). Example if we write matrix T2 =R2/Z2 for the flat torus, then the automorphism of R2 given by the matrix (1112) induces an Anosov diffeomorphism on T2. On the Klein bottle, however, it is impossible to construct an Anosov diffeomorphism. This raises the obvious question: On which manifolds can we construct Anosov diffeomorphisms? c.f. Smale [14] p.760. Smale gives examples of Anosov diffeomorphisms on nilmanifold (p. 761). Shub [13] gives examples on a fourdimensional flat manifold which is not a torus, and on a six dimensional infranil manifold. We give below a complete algebraic characterization of those flat manifolds whichsupport Anosov diffeomorphisms (see Theorem 2.3.1). Each flat manifold comes prepacked with its own finite group F (the linear holonomy group) and a representation T of this group into GL(n, Z), where '" n is the dimension of the manifold. In chapter 1 we find necessary and also sufficient conditions for M to support an Anosov diffeomorphism, and show that these depend only on 2. the representation T. In chapter 2 we examine those cconditions, as a problem in abstract representation theory and arrive at the surprising conclusion that the conditions are equivalent. They depend on the manner in which T decomposes as we enlarge the coefficient domain first from Z to Q and then to R. What we do is this : first we decompose T over. Q. If any pieces occur more than once in the decomposition we ignore them. We non take those pieces which occur precisely once and attempt to decompose them over R. If we are successful every time, the manifold will support an Anosov diffeomorphism, but if any of them is irreducible over R, then the manifold will not support an Anosov diffeomorphism. In chapter 3 we apply our results to specific problems, generate lots of examples and finally use one of the examples to illustrate a formula of Williams [15] on zeta functions of diffeomorphisms. To reduce the weight of the proofs in chapters 1 and 2, we have assembled those parts of the proofs which have nothing to do with Anosov diffeomorphisms into a chapter 0 which we call "Prerequisites". It is used heavily for reference, and to establish notation.
