The dynamical zeta function

Transcription

1 The dynamical zeta function JWR April 30, An invertible integer matrix A GL n (Z generates a toral automorphism f : T n T n via the formula f π = π A, π : R n T n := R n /Z n The set Fix(f := {x T n : f(x = x} of fixed points of f is the ernel of the homomorphism 1 f : T n T n. Now df(x = A for x T n and, if 1 df(x is invertible whenever f(x = x, then the fixed points of f are isolated and hence finite in number. In fact Theorem 2. If det(i A 0 then #Fix(f = det(i A. Lemma 3 (Smith Normal Form 1. If B Z m n there exist P GL m (Z and Q GL n (Z such that B = P DQ where D is diagonal, (i.e. D ij = 0 for i j and, where d i := D ii we have d i 0 and d i divides d i+1 for i = 1,..., min(m, n. Proof. Do row and column operations to get the gcd of the B ij in the (1, 1 position and zero elsewhere in the first row and column and then induct on the size of the matrix. Proof of Theorem 2. By Smith I A = P DQ so 1 f = φ δ ψ where φ and ψ are automorphisms and δ(π(x = π(d 1 x, d 2 x,..., d n x, 1 This is essentially the fundamental theorem for abelian groups. 1

2 Hence det(i A = det(p det(d det(q = ±d 1 d n. Now ( 1 x er(δ x = π,..., n d 1 d n for some integers i with 1 = 0,..., i 1. Because φ and ψ are automorphisms, # er(1 f = # er(δ. Hence det(i A = d1 d n = # er(δ = # er(1 f = #Fix(f Definition 4. Let f : X X have the property that N := #Fix(f is finite for all = 1, 2,.... Then the dynamical zeta function for f is the formal power series ( N t ζ f (t := exp. Note that each orbit of f or period consists of fixed points of f so that log ζ f (t is the generating function for the number N / of periodic orbits of period. Theorem 5. Let A {0, 1} n n and let σ A : Σ A Σ A be the corresponding subshift of finite type. Then ζ σa (t = det(i ta 1. Proof. The (i, j in A is the number of paths in the incidence graph from i to j Hence N = tr(a is the number of closed paths in the incidence graph, i.e. the number if distinct elements of Σ A which are periodic of period. Thus ( tr(a t ζ(t = exp = exp tr ( log(i ta = det exp( log(i ta = det(i ta 1 where we have used the formula exp tr(b = det exp(b (the exponential of the sum of the eigenvalues is the product of the exponentials of the eigenvalues. 2

3 Theorem 6. Assume that N = det(i A and define ζ(t = N t as above. Then. ζ(t = det ( I tλ j A ( 1 j+1. Proof. For any matrix B R n n we have det(i B = ( 1 j tr(λ j B where Λ j B : Λ j R n Λ j R n is the map on the jth exterior power induced by B : R n R n. Note that Λ j is functorial: Λ j (B 1 B 2 = (Λ j B 1 (Λ j B 2. Now t N log ζ(t = t = det(i A t = ( 1 j tr(λ j A ( = ( 1 j t (Λ j A tr = ( 1 j tr ( log(i tλ j A = = tr ( ( ( 1 j+1 tr log I tλ j A ( ( ( 1 j+1 log I tλ j A 3

4 so exponentiating ζ(t = det exp log ( I tλ j A ( 1 j+1 = det ( I tλ j A ( 1 j+1 7. Assume f : M M is a smooth map of a compact oriented manifold of dimension n and let Γ M M denote the graph of f and M M denote the diagonal, Then a point p M is a fixed point of f (i.e. f(p = p iff (p, p Γ iff p F 1 (. Then the submanifolds Γ and of M M intersect transversally at (p, p if and only if the I df(p : T p M T p M is invertible. In this case we say that the fixed point is nondegenerate and define the fixed point index of f at p by i(f, p := sgn det(1 df((p. Since ( I df(p 0 det(i df(p = det df(p I ( I I = det df(p I we see that i(f, p = 1 if the orientation of T (p,p Γ induced by the graph map and the orientation of T (p,p induced by the diagonal map add up to the orientation of T (p,p M M and that i(f, p = 1 otherwise. Theorem 8 (Lefschetz Fixed Point Theorem. Assume that f has only nondegenerate fixed points. Then f(p=p i(f, p = ( 1 j tr ( H j (f : H j (M, R H j (M, R. Proof. [1] page 129. Theorem 9. Let f be the toral automorphism induced by A GL n (Z and assume A is hyperbolic (no eigenvalue on the unit circle then there is an A invariant splitting R n = E s E u, an inner product on R n, and a number λ (0, 1 such that Av λ v for v E s and A 1 v λ v for v E u. Let ζ(t be defined in Theorem 6 and u = dim E u. Then ζ f (t = ζ(t ( 1u if A E u preserves orientation and ζ f (t = ζ(t ( 1u+1 if A E u reverses orientation. 4

5 Proof. For M = T n and f the toral automorphism induced by A we have H j (M, R = Λ j R n and H j (f = Λ j A. Since df(p = A the left hand side of the Lefschetz fixed point formula is ±#Fix(f where ± = sgn det(i A and the right hand side is det(i A = j ( 1j tr(λ j A. Now ± = ( 1 u if det(a E u > 0 and ± = ( 1 u+1 if det(a E u < 0. Now mimic the proof of Theorem 6 replacing t by (±t. Definition 10. A diffeomorphism f : M M is called Anosov iff there is an f invariant splitting T M = E s E u, a Riemannian metric on M, and a number λ (0, 1 such that df(pv λ v for v E s p and df 1 (pv λ v for v E u p. Theorem 11. The zeta function of an orientation preserving Anosov diffeomorphism on a compact orientable manifold is rational. Proof. If E u is orientable and df E u preserves orientation the proof is essentially the same as the proof of Theorem 6: read H j for Λ j and use that H j is functorial. The tricy part is to see that at a point p of period the index i(f, p is depends only on f and the parity of. 2 (See [5] and the other references. References [1] R. Bott & L. W. Tu: Differential Forms in Algebraic Topology [2] C. Robinson: Dynamical Systems, CRC Press, [3] M. Shub: Global Stability of Dynamical Systems, Springer [4] S. Smale: Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 ( repreinted in The Mathematics of Time, Springer- Verlag 1980, [5] R. Williams: The zeta function of an attractor, in Conference on the topology of manifolds (J.C. Hocing Ed, pp Prendle Weber Schmidt 2 Is this true? 5