Lecture 2: Ruelle Zeta and Prie Nuber Theore for Graphs Audrey Terras CRM Montreal, 2009 EXAMPLES of Pries in a Graph [C] =[e e 2 e 3 ] e 3 e 2 [D]=[e 4 e 5 e 3 ] e 5 [E]=[e e 2 e 3 e 4 e 5 e 3 ] e 4 ν(c)=3, ν(d)=3, ν(e)=6 e E=CD another prie [C n D], n=2,3,4, infinitely any pries
Ihara Zeta Function ζ V (u, X) = [C] pri e in X Ihara s Theore (Bass, Hashioto, etc.) ν (C) ( - u ) A = adjacency atrix of X ( V x V atrix of 0s and s ij entry is iff vertex i adjacent to vertex j) Q = diagonal atrix jth diagonal entry = degree jth vertex -; r = rank fundaental group = E - V + - 2 r- 2 ζ (u,x) = (-u ) det(i -Au + Qu ) For K 4 r= E - V +=6-4+=3 ζ 0 2 0 0 0 0 0 2 0 0 A=, Q = 0 0 0 2 0 0 0 0 0 2 ( ) 2 2 2 3 uk, = ( u) ( u)( 2 u)( + u+ 2 u) 4 - The Edge Matrix W Define W to be the 2 E 2 E atrix with i j entry if edge i feeds into edge j, (end vertex of i is start vertex of j) provided i opposite of j, otherwise the i j entry is 0. i j Theore. ζ(u,x) - =det(i- W u). Corollary. The poles of Ihara zeta are the reciprocals of the eigenvalues of W. Recall that R = radius of convergence of Dirichlet series for Ihara zeta. Note: R is closest pole of zeta to 0. The pole R of zeta is: R=/Perron-Frobenius eigenvalue of W. See Horn & Johnson, Matrix Analysis 2
Ruelle Zeta which ay also be called Dynaical Systes Zeta or Sale Zeta Reference: D. Ruelle, Dynaical Zeta Functions for Piecewise Monotone Maps of the Interval, CRM Monograph Series, Vol. 4, AMS, 994 T. Bedford, M. Keane, and C. Series, (Eds), Ergodic Theory, Sybolic Dynaics, and Hyperbolic Spaces Ruelle s otivation for his definition cae partially fro Artin and Mazur, Annals of Math., 8 (965). They based their zeta on the zeta function of a projective non-singular algebraic variety V of diension n over a finite field k with q eleents. If N is the nuber of points of V with coordinates in the degree extension field of k, the zeta function of V is: Z V N u ( u) = exp. N = Fix(F ), where F is the Frobenius ap taking a point with coordinates x i to the point with coordinates (x i ) q. 3
Weil conjectures, proved by Deligne, say ( ) where the P j are polynoials with zeros of absolute value q -j/2. Moreover the P j have a cohoological eaning as det(-uf* H j (V)). 2n j= 0 j+ ZV( u) = Pj( u), Artin and Mazur replace the Frobenius of V with a diffeoorphis f of a sooth copact anifold M - defining their zeta function u ζ ( u) = exp Fix( f ). = Ruelle zeta function Suppose M is a copact anifold and f:m M. Assue the following set finite: Fix(f )={x M f (x)=x}. st type Ruelle zeta is defined for atrix-valued function ϕ:m C dxd u k ζ ( u ) = exp Tr ( f ( x )) ϕ x Fix( f ) k= 0 A special case: ϕ= u ζ ( u) = exp Fix( f ). = I=finite non-epty pyset (our alphabet). For a graph X, I is the set of directed edges. The transition atrix t is a atrix of 0 s and s with indices in I. In the case of a graph X, t is the 0, edge atrix W defined earlier, which has i,j entry if edge i feeds into edge j (eaning that terinal vertex of I is the initial vertex of j) provided edge i is not the inverse of edge j. 4
Note: I Z is copact and so is the closed subset {( ξ k ) t k k Z ξξ } k k+ Λ= =,. In the graph case ξ Λ corresponds to a path without backtracking. A continuous function τ:λ Λ such that τ(ξ) k =ξ k+ is called a subshift of finite type. Prop.. (Bowen & Lanford). As the Ruelle zeta of a subshift of finite type, the Ihara zeta is the reciprocal of a polynoial: u ζ ( u) = exp Tr( t ) = I ut det( ). t=w for graphs Proof. By the first exercise below, we have the first equality in the theore. u ζ ( u) = exp Tr( t ) Then Fix(τ ) =Tr(t ). This iplies using the 2nd exercise: ζ(u)=exp(tr(-log(-ut)))=det(i-ut) -. Exercise A. Show that in the graph case, Fix(τ ) =the nuber of length closed paths without backtracking or tails in the graph X with t=w, fro our previous discussion of graphs. Exercise B. Show that exp(tra)=det(expa) for any atrix A. Hint: Use the fact that there is a non-singular atrix B such that BAB - =T is upper triangular. 5
Define R as the radius of convergence of the Ihara zeta. It is also the closest pole to 0. For a (q+)-regular graph, R=/q. For K 4, R=/2. Define Δ as the g.c.d. of the prie lengths. For K 4, Δ=. Theore. (Graph prie nuber theore) If the graph is connected and Δ divides π() = #{prie paths of length } ~ ΔR - /, as If Δ does not divide, π()=0. Proof. If N =# {closed paths C,length,no backtrack,no tails} we have Then ζ(u,x) - =det(i-w u) iplies dlog ζ (u,x) u = Nu du d log ζ (u) = N d u u = u λ log( u) du n λ Spec ( W ) du = λ u λ Spec ( W ) N = λ Spec(W ) λ = The doinant ters in this su are those coing fro the eigenvalues λ of W with λ =/R. 6
N = λ Spec(W ) λ Theore. (Kotani and Sunada using Perron- Frobenius Th.) The poles of ζ on u =R have the for Re 2πia/Δ, where a=,2,...,δ. By this theore, Δ 2πi Δ a - λ λ spec(w ) a= λ axial N = R e The su is 0 unless divides Δ, when it is ΔR -. Next we need a forula to relate N and π(). This iplies n ( ) π ζ (u,x) = -u Möbius inversion says n - (n) dlog ζ (u,x) u = d π(d)u du d N = d π(d) d π ( ) = μ ( N d ) d d Note that π()=0 for graph without loops. If =prie p Z, we have N p =pπ(p). Thus the prie nuber theore follows. Exercise. Fill in the details in this proof. 7
Tetrahedron or K 4 exaple d x log ζ X(x)= Nx dx = 24x 3 + 24x 4 + 96x 6 + 68x 7 + 68x 8 + 528x 9 + 200x 0 + 848x + 3960x 2 + 8736x 3 + 628x 4 + 3944x 5 + 66888x 6 +. 8 prie paths of length 3 on the tetrahedron. Check it! We count 4 plus their inverses to get 8. 6 prie paths of length 4. Check. 0 paths of length 5 6 paths of length 6. That is harder to check. Question: 528 is not divisible by 9. Shouldn't it be? Answer: You only know divides N when is prie. π( ) = μ( ) Nd d d Exercises. List all the zeta functions you can and what they are good for. There is a website that lists lots of the: www.aths.ex.ac.uk/~watkins 2. Copute Ihara zeta functions for the cube, dodecahedron, buckyball, your favorite graph. Matheatica or Matlab or Scientific Workplace etc. should help. 3. Do the basic graph theory exercise 4 on page 24 of the anuscript on y website: www.ath.ucsd.edu/~aterras/newbook.pdf 4. Show that the radius of convergence of the Ihara zeta of a (q+)-regular graph is R=/q. Explain why the closest pole of zeta to the origin is at R. 5. Prove the functional equations of Ihara zeta for a regular graph. See p. 25 of y anuscript. 6. Look up the paper of Kotani and Sunada and figure out their proof. You need the Perron Frobenius theore fro linear algebra. [Zeta functions of graphs, J. Math. Soc. Univ. Tokyo, 7 (2000)]. 8
6. Fill in the details in the proof of the graph theory prie nuber theore. 7. Prove the prie nuber theore for a (q+)-regular graph using Ihara s theore with its 3-ter deterinant rather than the /det(i-uw ) forula. 8. Exercise 6 on page 28 of y anuscript. This is a Matheatica exercise to plot poles of Ihara zetas. 9. Show that in the graph case, Fix(τ ) =the nuber of length closed paths without backtracking or tails in the graph X with t=w, fro our previous discussion of graphs. 0. Show that exp(tra)=det(expa) for any atrix A. Hint: Use the fact that there is a non-singular atrix B such that BAB - =T is upper triangular. 9