By J. W. S. Cassels, A. Frohlich
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6) was first proved by Smale [S51. It said f is isotopic to a diffeomorphism with hyperbolic chain recurrent set and zero dimensional basic sets. Shub, Smale, and Williams added the strong transversality condition to this. The version of the theorem given here appeared subsequently in [SS). 7) and several of the results of Appendix fi for the case of nonsimply connected mani- folds can be found in [Mal]. 7) show the close connection between the dynamics on the basic sets and homological invariants.
Let h(t) = det(I - g, , t), and note g' , preserves the nonsingular intersection pairing ( , ): H1(N; F2) x H1(N; F2) - F2. IfJ is a matrix for this pairing and A a matrix for g,, then ArJA = J so A' = JA-'J"' . Hence h(t'') = det(I - At -') = det(A -'t)det(A -'t - I) = t kh(t). Since det(I - g*,t) and det(I -g,2t) both have the form II(1 + tp'), it follows that p(t -1) = t'p(t) for some r E Z. We sketch the proof of the converse, supposing first that p(t) has even degree. Then choose a diffeomorphism f: M2 _M2 such that p(t) = det(I - f,t t) HOMOLOGY AND DYNAMICAL SYSTEMS 47 where ffj: HI(M2; F2) -III(M2; F2).
Suppose that A is a basic set of a surface diffeomorphism f: M2 _ M2 which has hyperbolic chain recurrent set. Then there exists a diffeomorphism g: N - N of a closed surface such that (a) The diffeornorphism g has hyperbolic chain recurrent set. (b) The only basic set of index I of g is A' and f (A is topologically conjugate to g(A'. (c) All other basic sets of g have index 0 or 2, and hence are periodic sources or sinks. PROOF. , In M - Mi. Let X = cl(M, - f(M,)). The Euler characteristic X(X) = X(M,) - XU(MI)) = 0.
Algebraic number theory Proc by J. W. S. Cassels, A. Frohlich