By Bowen J.

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1) As in the last chapter we begin with an example. 1). What is shown is a picture of T2 with a disk D2 deleted and the image of T2 - D2 under a diffeomorphism f: T2 --+ T2. 10)). The geometric intersection matrix for H with respect to f is A = (0, i ). Thus if A= n;%__ A is a compact invariant hyperbolic set and JhA is topologically conjugate to o(A): EA -- EA. o f"(Mo) = {p} and l">o f-"(D2) = {00} where D2 = cl(M2 -Mr). Since 27 28 JOHN M. FRANKS the chain recurrent set R is invariant it follows that R n Mo = (p), and R n D2 = -, so R C (p.

If a matrix A is not irreducible we can still describe the chain recurrent set of a(A). For example if A = (o where B and C are in x m and n x n irreducible matrices respec. tively, then c) . X - {c E EA 11 < r(ct) < m, 1 < 1(ct) < in, for all I) is a cloned invariant set and o(A)IX is topologically conjugate to o(B). Likewise there is a closed invariant set Y with o(A)I Y conjugate to a(C). Consjderation of the edge graph associated to A shows that W represents nonrecurrent orbits going from Y to X.

1 if orientation is reversed. (fm) where this sum is over all x E Fix(fm). Let K be as above and define p = exp(E(l/m)L,,,(K)tm) where Lm(K) is the sum of I1(fm) for all x E Fix(f m) ( K. Then form < n we have L,,, (K) = L(f m) since all points of period < n are in K. Thus the coefficient of t" in p is the same as the coefficient of t" in Z(f). ) where L,,(-y,) = EIJ(fm); the sum being taken over x E y1 rl Fix(fm). Hence p = jj exp[ F_ MLm(71)tml . 1=1 m=1 / Since Lm(7) = 0 if mit Omodp(y) P(7X-I)u(-Y)A-lp(7) if m = 0 mod p(7).