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Zeta functions for two-dimensional shifts of finite type

Wen-Guei Hu

Shing-Tung Yau Center National Chiao Tung University, Hsinchu

April 26, 2015

Workshop on Combinatorics and Applications at SJTU

(Joint work with Prof. Jung-Chao Ban, Prof. Song-Sun Lin and

• Dr. Yin-Heng Lin.)

Introduction Main results Ising model Further remarks

Jung-Chao Ban, Wen-Guei Hu, Song-Sun Lin and Yin-Heng Lin, Zeta functions for two-dimensional shifts of finite type, Memoirs of the American Mathematical Society,

• Vol. 221, No. 1037 (2013).

Wen-Guei Hu and Song-Sun Lin, Zeta functions for higher-dimensional shifts of finite type, International J. of Bifurcation and Chaos, Vol. 19, No. 11 (2009) 3671-3689.

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks

Outline

1

Introduction Riemann zeta function 1-d dynamical zeta function Thermodynamic zeta function Dynamical zeta function for Zd-actions

2

Main results Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

3

Ising model

4

Further remarks

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Riemann zeta function 1-d dynamical zeta function Thermodynamic zeta function Dynamical zeta function for Zd-actions

Outline

1

Introduction Riemann zeta function 1-d dynamical zeta function Thermodynamic zeta function Dynamical zeta function for Zd-actions

2

Main results Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

3

Ising model

4

Further remarks

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Riemann zeta function 1-d dynamical zeta function Thermodynamic zeta function Dynamical zeta function for Zd-actions

Riemann zeta function

(1) Riemann zeta function: ζ(s) :=

∞

• n=1

n−s. (1) Euler product formula:

ζ(s) =

• p:prime
• 1 − p−s−1 .

(2)

Meromorphy:

Riemann showed that ζ(s) can be extended meromorphically to C with a single pole at s = 1.

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Riemann zeta function 1-d dynamical zeta function Thermodynamic zeta function Dynamical zeta function for Zd-actions

Riemann zeta function

Functional equation:

relation between ζ(s) and ζ(1 − s).

Location of zeros:

Riemann hypothesis: all nontrivial zeros are on the line Re(s) = 1

2.

Asymptotic formula:

the number of primes up to x is ∼

x log x. Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Riemann zeta function 1-d dynamical zeta function Thermodynamic zeta function Dynamical zeta function for Zd-actions

Outline

1

Introduction Riemann zeta function 1-d dynamical zeta function Thermodynamic zeta function Dynamical zeta function for Zd-actions

2

Main results Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

3

Ising model

4

Further remarks

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Riemann zeta function 1-d dynamical zeta function Thermodynamic zeta function Dynamical zeta function for Zd-actions

(2) Artin-Mazur zeta function (1965) (Dynamical zeta function)

[ M. Artin and B. Mazur, On periodic points, Annals Math. 81 (1965), 82-99.]

φ : X → X homeomorphism on compact spaces. Γn(φ): the number of fixed point of φn. ζφ(s) := exp ∞

• n=1

Γn(φ) n sn

• .

(3) zeta function is defined only if Γn(φ) < ∞ for all n ≥ 1.

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Riemann zeta function 1-d dynamical zeta function Thermodynamic zeta function Dynamical zeta function for Zd-actions

Product formula:

ζφ(s) =

• γ
• 1 − s|γ|−1

, (4) where the product is taken over all periodic orbits γ of φ and |γ| denotes the number of points in γ.

Bowen and Lanford (1970)

[ R. Bowen and O. Lanford, Zeta functions of restrictions of the shift transformation, Proc. AMS Symp. Pure Math. 14 (1970), 43-49.]

Theorem: If φ is a shift of finite type, then ζφ is a rational function.

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Riemann zeta function 1-d dynamical zeta function Thermodynamic zeta function Dynamical zeta function for Zd-actions

1-dim shifts of finite type

Color set Sp = {0, 1, · · · , p − 1}, p ≥ 2 Basic set of admissible local patterns B ⊂ SZ2×1

p

Σ(B): the set of all global patterns on Z1 that can be generated by B Pn(B), n ≥ 1: the set of all n-periodic patterns that can be generated by B, i.e., (xi)∞

i=−∞ ∈ Σ(B) with

xj = xj+n for all j ∈ Z. x1 x1 x2 x2 xn xn Pn(B) = ♯Pn(B)

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Riemann zeta function 1-d dynamical zeta function Thermodynamic zeta function Dynamical zeta function for Zd-actions

Example: (Golden-Mean shift)

Basic set of admissible local patterns: BG =

• ,

1

,

1

• Transition matrix AG =

1 1 1

• P1(BG) = {0∞} → P1(BG) = 1 = tr(AG)

P2(BG) = {(00)∞, (01)∞, (10)∞} → P2(BG) = 3 = tr(A2

G)

. . . ⇒ Pn(BG) = tr (An

G)

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Riemann zeta function 1-d dynamical zeta function Thermodynamic zeta function Dynamical zeta function for Zd-actions

Eigenvalues of AG: g = 1+

√ 5 2

, ¯ g = 1−

√ 5 2

− log(1 − s) =

∞

• k=1

sk k

Then, ζAG(s) ≡ exp ∞

• k=1

Pn(AG) k

sk

• =

exp ∞

• k=1

tr(Ak

G)

k

sk

• =

exp ∞

• k=1

gk+¯ gk k

sk

• =

1 (1−gs)(1−¯ gs)

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Riemann zeta function 1-d dynamical zeta function Thermodynamic zeta function Dynamical zeta function for Zd-actions

ζAG(s) =

• γ
• 1 − s|γ|−1

, (5)

where the product is taken over all periodic orbits γ of φ and |γ| denotes the number of points in γ.

Example:

|γ| = 1 → γ = {0∞}, |γ| = 2 → γ = {(01)∞, (10)∞} = {(01)∞, σ ((01)∞)}, |γ| = 3 → γ = {(001)∞, (010)∞, (100)∞} = {(001)∞, σ ((001)∞) , σ2 ((001)∞)} Then, ζAG(s) =

• γ
• 1 − s|γ|−1

= 1 1 − s · 1 1 − s2 · 1 1 − s3 · · · · (6)

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Riemann zeta function 1-d dynamical zeta function Thermodynamic zeta function Dynamical zeta function for Zd-actions

A: m × m transition matrix. ζA(s) := exp ∞

• k=1

tr(Ak) k sk

• = (det(I − sA))−1

=

• λ∈Σ(A)

(1 − λs)−χ(λ), (7) χ(λ): algebraic multiplicity.

∞

• k=1

Ak k sk = log(I − sA)−1

(8) exp (tr(M)) = det (exp(M)) (9)

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Riemann zeta function 1-d dynamical zeta function Thermodynamic zeta function Dynamical zeta function for Zd-actions

Outline

1

Introduction Riemann zeta function 1-d dynamical zeta function Thermodynamic zeta function Dynamical zeta function for Zd-actions

2

Main results Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

3

Ising model

4

Further remarks

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Riemann zeta function 1-d dynamical zeta function Thermodynamic zeta function Dynamical zeta function for Zd-actions

(3) Ruelle: Thermodynamic zeta function (1978)

[ D. Ruelle, Thermodynamic Formalism, Addison-Wesley, 1978.]

ζR(s) := exp ∞

• n=1

Zn(θ, α) n sn

• ,

(10)

where Zn(θ, α) =

• x∈Fixαn
• exp

n−1

• k=0

θ(αkx)

• (11)

is a partition function with periodic boundary conditions.

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Riemann zeta function 1-d dynamical zeta function Thermodynamic zeta function Dynamical zeta function for Zd-actions

Outline

1

Introduction Riemann zeta function 1-d dynamical zeta function Thermodynamic zeta function Dynamical zeta function for Zd-actions

2

Main results Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

3

Ising model

4

Further remarks

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Riemann zeta function 1-d dynamical zeta function Thermodynamic zeta function Dynamical zeta function for Zd-actions

(4) J.C. Ban, S.S. Lin and Y.H. Lin (2005): Zeta functions for 2-d shifts of finite type. Basic lattice: Z2×2 Set of symbols / colors: Sp = {0, 1, · · · , p − 1}. In particular, S2 = {0, 1} = { , } Σ2×2(p) := SZ2×2

p

: the set of all local patterns. Basic admissible set B: B ⊂ Σ2×2(p).

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Riemann zeta function 1-d dynamical zeta function Thermodynamic zeta function Dynamical zeta function for Zd-actions

Idea: 2-d periodic patterns with horizontal period n and vertical

period k, n, k ≥ 1 PB

• n

k

• = {(xi,j ∈ Σ(B))

| xi+n,j = xi,j+k = xi,j for i, j ∈ Z}

x1,1 x1,1 x1,1 x1,1 x2,1 x2,1 xn,1 xn,1 x1,2 x1,2 x2,2 xn,2 x1,k x1,k x2,k xn,k

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Riemann zeta function 1-d dynamical zeta function Thermodynamic zeta function Dynamical zeta function for Zd-actions

Define ζ♯

B(s) ≡ exp

∞

• n=1

∞

• k=1

1 nk ΓB n k

• snk
• ,

(12)

where ΓB n k

• = ♯PB

n k

• .

Then, ζ♯

B(s) = ∞

• n=1

exp

• 1

n

∞

• k=1

1 k tr(Tk

n(B))snk

• .

(13)

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Riemann zeta function 1-d dynamical zeta function Thermodynamic zeta function Dynamical zeta function for Zd-actions

However, ζ♯

B(s) is not (an infinite product of) rational functions.

Example 1: Consider B0 =

• .

Clearly, ΓB0

• n

k

• = 1 for all n, k ≥ 1.

Then, ζ♯

B0(s) = ∞

• n=1

exp

• 1

n

∞

• k=1

1 k snk

• =

∞

• n=1

1 (1 − sn)1/n is not (an infinite product of) rational functions.

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Riemann zeta function 1-d dynamical zeta function Thermodynamic zeta function Dynamical zeta function for Zd-actions

Dynamical zeta function for Zd-actions

(5) D. Lind: A zeta function for Zd-actions. (1996)

[ London Math. Soc. Lecture Note Series 228. Cambridge Univ. Press (1996) 433-450]

α: Zd-action Ld: the set of finite-index subgroups (lattices) of Zd. [L]: index

• Zd/L
• .

ΓL(α): the number of fixed points of αn for all n ∈ L. ζα(s) = exp  

L∈Ld

ΓL(α) [L] s[L]   . (14)

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Riemann zeta function 1-d dynamical zeta function Thermodynamic zeta function Dynamical zeta function for Zd-actions

Results (Lind):

(i) Analyticity: ζα(s) has radius of convergence exp(−g(α)), where

g(α) = lim sup

[L]→∞

log ΓL(α) [L] (15) is the growth rate of periodic points.

(ii) Product formula: For d = 2,

ζα(s) =

• γ

∞

• k=1

1 1 − s|γ|k

• ,

(16) where the product is over all finite orbits γ of α and |γ| denotes the number of points in γ.

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Riemann zeta function 1-d dynamical zeta function Thermodynamic zeta function Dynamical zeta function for Zd-actions

(iii) The Taylor series for ζα(s) has integer coefficients. Example 1: (continued)

ζB0(s) =

∞

• n=1

1 1 − sn =

∞

• n=0

p(n)sn, (17) where p(n) is the number of partitions of n.

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Riemann zeta function 1-d dynamical zeta function Thermodynamic zeta function Dynamical zeta function for Zd-actions

Open problems (Lind):

(1) For "finitely determined" Zd-actions α such as shifts of finite type, is there a reasonable finite description of ζα(s)? (2) Compute explicitly the thermodynamic zeta function for the 2-dimensional Ising model, where α is the Z2 shift action

• n the space of configurations.

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Riemann zeta function 1-d dynamical zeta function Thermodynamic zeta function Dynamical zeta function for Zd-actions

Thermodynamic zeta function for Zd-actions

Thermodynamic zeta function with weight function θ : X → (0, ∞): ζ0

α,θ(s) = exp

 

L∈Ld

  

• x∈fixL(α)
• k∈Zd/L

θ

• αkx
• 

  s[L] [L]   , (18) where fixL(α) is the set of points fixed by αn for all n ∈ L.

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

Outline

1

Introduction Riemann zeta function 1-d dynamical zeta function Thermodynamic zeta function Dynamical zeta function for Zd-actions

2

Main results Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

3

Ising model

4

Further remarks

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

Hermite normal form

Lind’s formulation:

L2 = a b c d

• Z2

ad − bc ≥ 1, a, b, c, d ∈ Z

• .

L2 can be parameterized by using Hermite normal form: L2 =

• n

l k

• Z2 : n ≥ 1, k ≥ 1 and 0 ≤ l ≤ n − 1
• =
• k

l n

• Z2 : n ≥ 1, k ≥ 1 and 0 ≤ l ≤ n − 1
• .

(19) n l k

• Z2

x y n k l k l n

• Z2

x y n k l

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

Hermite normal form

ΓB

• n

l k

• : the number of
• n

l k

• periodic and

B-admissible patterns on Z2. Lind’s zeta function (11) ⇒ ζ0

B(s) =

exp ∞

• n=1

∞

• k=1

n−1

• l=0

1 nkΓB

n l k

• snk
• .

(20)

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

Double sum Iterated sums For n ≥ 1, n-th order zeta function is defined by ζB;n(s) ≡ exp

• 1

n

∞

• k=1

n−1

• l=0

1 k ΓB n l k

• snk
• .

(21) The zeta function ζB(s) ≡

∞

• n=1

ζB;n. (22)

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

Trace operator and reduced trace operator

J.C. Ban, S.S. Lin and Y.H. Lin (2007)

[ J.C. Ban, S.S. Lin and Y.H. Lin, Patterns generation and spatial entropy in two dimensional lattice models, Asian J. Math. 11 (2007), no. 3, 497–534.]

ψn: counting function for local patterns (β1, β2, · · · βn, β1). ψn(β1, β2, · · · βn, β1) = 1 +

n

• j=1

βj2n−j, where βi,j ∈ {0, 1}.

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

Cylindrical matrices C2×2 = (23)

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

= (24) x-periodic of period 2 with height 2.

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

Cn×2: x-cylindrical matrix of order n. : x-periodic of period n with height 2. Cn×2 =        β1,2 β2,2 · · · · · · βn,2 β1,2 β1,1 β2,1 βn,1 β1,1       

2n×2n

(25) where βi,j ∈ {0, 1}. Cn×2 is rotationally symmetric.

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

Trace operator Tn

Trace operators Tn : transition matrix of patterns which are

B-admissible and x-periodic of period n with height 2.        β1,2 β2,2 · · · · · · βn,2 β1,2 β1,1 β2,1 βn,1 β1,1       

2n×2n

.

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

Cn×2 is rotationally symmetric. rotation β1 β2 σ(β1) σ(β2) β1,1 β1,1 β2,1 β2,1 β3,1 βn,1 ⇒ Proposition 2.1 Tn is rotationally-symmetric: Tn;σl(i),σl(j) = Tn;i,j 0 ≤ l ≤ n − 1, 1 ≤ i, j ≤ 2n.

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

Rotational matrix Rn

σn: shift to left. Rotational matrix Rn (2n × 2n): β1 β1 β1 β2 β2 β2 · · · · · · βn βn Rn σn i j = σn(i) ψn ψn Rn ≡

n−1

• l=0

Rl

n. Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

Example 2:

R2=     1 1 1 1    , and        C2(1) = {1}, ω2,1 = 1, C2(2) = C2(3) = {2, 3}, ω2,2 = 2, C2(4) = {4}, ω2,4 = 1, I2 = {1, 2, 4}. 1 1 1 1 1 2 3 σ2 σ2 R2 R2 ψ2 ψ2 ψ2 ψ2

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

1 1 2 3 1 1 1 5 2

σ3 σ3 σ3 R3 R3 R3 R3 R3 R3 ψ3 ψ3 ψ3 ψ3

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

Number of periodic patterns Trace of trace operators

Theorem 2.2 Given B ⊂ Σ2×2, for any 0 ≤ l ≤ n − 1, k ≥ 1, ΓB n l k

• = tr
• Tk

nRl n

• and

n−1

• l=0

ΓB n l k

• = tr
• Tk

nRn

• ,

where Rn =

n−1

• l=0

Rl

n. Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

Trace operator Tn(B)

Example:

3 2

• periodic

↔ diag(C2

3×2) x

u1,1 u1,1 u1,1 u1,1 u2,1 u2,1 u3,1 u3,1 u3,1 u3,1 u1,2 u1,2 u1,2 u1,2 u1,2 u1,2 u2,2 u2,2 u2,2 u3,2 u3,2 u3,2 u3,2 u3,2 u3,2

ΓB 3 2

• = tr T2

3

.

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

Trace operator Tn(B)

3 1 2

• periodic

↔ diag(C2

3×2R3) x

u1,1 u1,1 u1,1 u1,1 u2,1 u2,1 u2,1 u2,1 u3,1 u3,1 u3,1 u3,1 u1,2 u1,2 u1,2 u1,2 u1,2 u1,2 u2,2 u2,2 u2,2 u2,2 u2,2 u2,2 u3,2 u3,2 u3,2 u3,2 u3,2 u3,2 ΓB 3 1 2

• = tr
• T2

3R3

• .

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

Trace operator Tn(B)

3 2 2

• periodic

↔ diag(C2

3×2R2 3) x

u1,1 u1,1 u1,1 u1,1 u1,1 u2,1 u2,1 u2,1 u2,1 u3,1 u3,1 u3,1 u3,1 u3,1 u1,2 u1,2 u1,2 u1,2 u1,2 u1,2 u1,2 u2,2 u2,2 u2,2 u2,2 u2,2 u2,2 u2,2 u3,2 u3,2 u3,2 u3,2 u3,2 u3,2 u3,2 ΓB 3 2 2

• = tr
• T2

3R2 3

• .

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

Rationality of ζn

Theorem 2.2 ⇒ ζB;n(s) = exp

• 1

n

∞

• k=1

1 ktr

• Tk

nRn

• snk
• .

(26)

Theorem 2.3 For any n ≥ 1 ζB;n(s) =

• λ∈Σ(Tn)

(1 − λsn)−χ(λ), where χ(λ) is the number of linear independent symmetric eigenvectors and generalized eigenvectors of Tn with respect to eigenvalue λ.

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

An eigenvector U of Tn is (Rn-)symmetric if Rl

nU = U

for any 0 ≤ l ≤ n − 1. (27) U is anti-symmetric if

n−1

• l=0

Rl

nU = 0.

(28) Example: Let T2 =     1 1 1 1    .

Eigenvectors of T2: V1 =     1    , V2 =     1 1    , V3 =     1 −1    , V4 =     1    .

⇒ V1, V2, V4 are symmetric and V3 is anti-symmetric.

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

By the rotational symmetry of Tn(B), Lemma 2.4 For n ≥ 1, if TnU = λU, (29) then Tn(Rl

nU) = λRl nU

(30) for any 0 ≤ l ≤ n − 1. Furthermore, if U is a generalized eigenvector, then Rl

nU is

also a generalized eigenvector for any 0 ≤ l ≤ n − 1.

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

Sketch proof of Theorem 2.3

Assume Tn is symmetric, the Jordan canonical form of Tn is Tn = UJnUt, where U = (U1, · · · , UN) is a N × N matrix of linear independent eigenvectors Uj, 1 ≤ j ≤ N = 2n. Jn = diag(λj).

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ 1 n ∞

• k=1

1 ktr(Tk nRn)snk

=

1 ntr

• U

∞

• k=1

1 kJk nsnk

• UtRn
• =

1 ntr

• U log(I − Jnsn)−1UtRn
• =

N

• j=1

1 n

• Rn ◦ UjU t

j

• log(1 − λjsn)−1

=

N

• j=1

log(1 − λjsn)−χn(λj). χn(λj) ≡ 1 n

• Rn ◦ UjU t

j

• where ◦ is the Hadamard product , i.e.,

A = [ai,j] and B = [bi,j] ⇒ A ◦ B = [ai,jbi,j].

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Introduction Main results Ising model Further remarks Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

Lemma 2.5 Let U = (u1, · · · , uN)t and W = (w1, · · · , wN), N = 2n. If U is symmetric, then 1 n |Rn ◦ U W | =

N

• j=1

ujwj. (31)

• if U = Uj and W = U t

j

⇒ 1

• (32)

If U is anti-symmetric, then 1 n |Rn ◦ U W | = 0. (33)

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ For n ≥ 2, Qn =                    

1 √n 1 √n 1 √n

· · ·

1 √n 1 √n

• n−1

n

−

1

√

n(n−1)

−

1

√

n(n−1)

· · · −

1

√

n(n−1)

−

1

√

n(n−1)

• n−2

n−1

−

1

√

(n−1)(n−2)

· · · −

1

√

(n−1)(n−2)

−

1

√

(n−1)(n−2)

. . . . . . . . . . . . . . . · · ·

1 √ 2

− 1

√ 2

                   

In particular, Q2 =  

1 √ 2 1 √ 2 1 √ 2

− 1

√ 2

 .

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Introduction Main results Ising model Further remarks Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

R(U) = Rl

nU

: 0 ≤ l ≤ n − 1 Lemma 2.6 For n ≥ 2, given an eigenvector U, define U1 = 1 √n

n−1

• l=0

Rl

nU

(34) and Uj =

• n − j + 1

n − j + 2Rj−2

n

U − 1 √n − j + 1√n − j + 2

n−1

• k=j−1

Rk

nU

(35) for 2 ≤ j ≤ n. Then, (i) if R(U) has rank κ, 1 ≤ κ ≤ n, then

• Uj

n

j=1 also has rank κ;

(ii) U1 is symmetric and Uj is anti-symmetric for 2 ≤ j ≤ n.

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

Example: Let U =

    1    . Then, R2U =     1    . ⇒ U 1 =

1 √ 2

    1     +

1 √ 2

    1     =

1 √ 2

    1 1    : symmetric U 2 =

1 √ 2

    1     − 1

√ 2

    1     =

1 √ 2

    1 −1    : anti-symmetric

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

Eλ: the eigenspace of Tn with eigenvalue λ. ⇒ Eλ can be spanned by linearly independent symmetric eigenvectors U1, U2, · · · , Up and anti-symmetric eigenvectors U ′

1, U ′ 2, · · · , U ′ p′, where p + p′ = dim(Eλ) and p

• r p′ may be zero.

⇒

χ(λ) = p = the number of linearly independent symmetric eigenvectors

• f Tn with eigenvalue λ.

The proof is complete.

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

Cn(i) = {σj(i)|0 ≤ j ≤ n − 1}: equivalent class. ωn,i is the cardinal number of Cn(i) . In = {i|1 ≤ i ≤ 2n, i ≤ σq(i), 1 ≤ q ≤ n − 1}.        C2(1) = {1}, ω2,1 = 1, C2(2) = C2(3) = {2, 3}, ω2,2 = 2, C2(4) = {4}, ω2,4 = 1, I2 = {1, 2, 4}, χ2 = 3.

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

Reduced trace operator τn

Definition 2.7 For n ≥ 1, Tn = [tn;i,j]. For each i, j ∈ In, define τn;i,j =

• k∈Cn(j)

tn;i,k (36) and denote the reduced trace operator of Tn by τn = [τn;i,j], which is a χn × χn matrix, where χn =

d|n

φ(d)2n/d. Here, φ(d) is the Euler totient function. χn is the number of different necklaces that can be made from n beads of two colors, when the necklaces can be rotated but not turned over.

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

Example 3: (2-d Golden-Mean Shift)

Forbidden set F =

• 1

1

,

1 1

• ⇒

H2 =     1 1 1 1 1 1 1     ⇒ T2 =     1 1 1 1 1 1 1     → τ2 =   1 2 1 1  

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

Theorem 2.8 (Rationality of ζn) Given B ⊂ Σ2×2(p), for any n ≥ 1, ζB;n(s) =

• λ∈Σ(Tn)

(1 − λsn)−χn(λ), (37) where χn(λ) is the number of linear independent symmetric eigenvectors and generalized eigenvectors of trace operator Tn with respect to eigenvalue λ, and ζB(s) =

∞

• n=1

ζB;n(s). (38) Furthermore, ζB;n(s) = (det (I − snτn))−1 (39) and ζB(s) =

∞

• n=1

(det (I − snτn))−1 , (40) where τn is the reduced trace operator.

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

Outline

1

Introduction Riemann zeta function 1-d dynamical zeta function Thermodynamic zeta function Dynamical zeta function for Zd-actions

2

Main results Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

3

Ising model

4

Further remarks

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

Zeta functions presented in skew coordinates

Recall GL2(Z) =

• a

b c d

• a, b, c, d ∈ Z and ad − bc = ±1
• .

Z2 is known to be invariant with respect to unimodular transformation. Given L = n l k

• Z2 ∈ L2, define

Lγ =

• n

l k

• γ

Z2 ≡ γt

• n

l k

• Z2.

L → Lγ is a bijection form L2 to itself, i.e., L2 = n l k

• γ

Z2 : n ≥ 1, k ≥ 1, 0 ≤ l ≤ n − 1

• .

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

For example, γ = 1 1 1

• Wen-Guei Hu

Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

The n-th order zeta function of ζ0

B(s) with respect to γ is defined

by ζB;γ;n(s) = exp

• 1

n

∞

• k=1

n−1

• l=0

1 k ΓB n l k

• γ
• snk
• .

(41) The zeta function ζB;γ with respect to γ is defined by ζB;γ(s) ≡

∞

• n=1

ζB;γ;n(s). (42)

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

Zeta functions presented in skew coordinates

Theorem 2.9 For any B ⊂ Σ2×2(p) and γ = a b c d

• ∈ GL2(Z),

ζB;γ;n(s) = (det (I − snτγ;n))−1 , (43) and ζB;γ(s) =

∞

• n=1

(det (I − snτγ;n))−1 . (44)

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

Zeta functions presented in skew coordinates

Theorem 2.10 For any admissible set B ⊂ Σ2×2(p) and γ ∈ GL2(Z), ζ0

B(s) = ζB;γ(s)

(45) for |s| < exp(−g(B)). Moreover, ζB;γ has the same (integer) coefficients in their Taylor series around s = 0.

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

Zeta functions presented in skew coordinates

Remark 2.11 For any B ⊂ Σ2×2(p), there is a family of zeta functions {ζB;γ|γ ∈ GL2(Z)}. The identity (29) yields a family of identities when ζB;γ is expressed as a Taylor series at the origin s = 0. These identities may have some interesting applications in number theory. Example 4: Let BI =

• ,

, ,

• .

⇒ χn = 1 n

• d|n

φ(d)2n/d = 1 n

n

• l=1

2(n,l).

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

Outline

1

Introduction Riemann zeta function 1-d dynamical zeta function Thermodynamic zeta function Dynamical zeta function for Zd-actions

2

Main results Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

3

Ising model

4

Further remarks

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

Meromorphicity of ζB;γ Natural boundary:

Let B ⊂ Σ2×2 and γ ∈ GL2(Z)

The meromorphic domain MB;γ of ζB;γ is defined by MB;γ = {s ∈ C|ζB;γ(s) is meromorphic at s}. The pole set PB;γ of ζB;γ is defined by PB;γ = {s ∈ C|1 − λsn = 0, where λ ∈ Σ(τB;γ;n) and n ≥ 1}. ζB;γ has a natural boundary ∂MB;γ if every point in ∂MB;γ is singular.

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

Meromorphicity of ζB;γ

Define λ∗

B;γ ≡ lim sup n→∞

 

• λ∈Σ(Tγ;n)

|λ|χγ;n(λ)  

1/n

. Let S∗

B;γ ≡

• λ∗

B;γ

−1 .

Theorem 2.12 Given an admissible set B ⊂ Σ2×2(p) and γ ∈ GL2(Z). Then zeta function ζB;γ is meromorphic in |s| < S∗

B;γ and may have poles in

PB;γ ∩

• s ∈ C| |s| < S∗

B;γ

• , i.e.,
• s ∈ C| |s| < S∗

B;γ

• ⊂ MB;γ.

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

Example 4: (continued)

According to the recursive formulas for Tn in [BLL],

Tn =        1 · · · 1 · · · . . . . . . · · · 1 · · · 1       

2n×2n

→ τn =        1 · · · 1 · · · . . . . . . · · · 1 · · · 1       

χn×χn

. Therefore, ζ(s) =

∞

• n=1

(1 − 2sn)−1.

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Introduction Main results Ising model Further remarks Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

On the other hand, consider ˆ γ =

• 1

1

• .

It can be shown that Tγ;n = I2n → τγ;n = Iχn Therefore, ζˆ

γ(s) = ∞

• n=1

(1 − sn)−χn.

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

(i) ζ(s) is meromorphic in |s| < 1 with poles

• 2−1/ne2πij/n : 0 ≤ j ≤ n − 1, n ≥ 1
• .

which is dense in S1 = {s ∈ C : |s| = 1})

⇒ natural boundary of ζ(s) is S1.

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Introduction Main results Ising model Further remarks Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

(ii) ζˆ

γ(s) is analytic in |s| < 1 2, since lim n→∞(χn)

1 n = 2.

(iii) ζ(s) = ζˆ

γ(s) in |s| < 1 2 ⇒

ζ(s) and ζˆ

γ(s) have the same (integer) coefficients in their Taylor

series at s = 0. ⇒ χn = 1 n

• d|n

φ(d)2n/d = 1 n

n

• l=1

2(n,l). (46)

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

Example 5: Consider

H2 =     1 1 1 1 1 1 1 1 1     . Then, V2 =     1 1 1 1 1 1 1 1 1     = G ⊗ G, where G = 1 1 1

• (47)

is the one-dimensional golden-mean matrix.

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Introduction Main results Ising model Further remarks Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

For any n ≥ 2, Tn = G⊗G ⊗ · · · ⊗ G⊗

• n−1 times⊗

G =

n−1

⊗ G, which is the n − 1 times Kronecker product of G. The spectrum of Tn is Σ(Tn) = {gn−j · gj|0 ≤ j ≤ n}, which has n + 1 members. The number of linear independent symmetric eigenvectors of gn−j · gj is χn,j = 1 n

• d|(j,n−j)

φ((j, n − j)/d)Cnd/(j,n−j)

jd/(j,n−j) ,

where φ is the Euler totient function.

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

Therefore, ζn(s) =

n

• j=0
• 1 − gn−jgjsn−χn,j

and ζ(s) =

∞

• n=1

ζn(s). It can be verified that lim sup

n→∞

max

0≤j≤n

• gn−jgjχn,j
• 1

n = 2,

which implies S∗ = 1

2 .

ζ(s) is analytic in |s| < 1

2. Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

• T2 = H2 ◦

V2 = H2 After the zero rows and columns have been deleted, Tn is a λn × λn full matrix, where

• λn+1 =

λn + λn−1 with

• λ2 = 3 and

λ3 = 4. Therefore,

• ζn(s) = (1 −

λnsn)−1 and

• ζ(s) =

∞

• n=1

(1 − λnsn)−1.

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks Rationality of ζn Zeta functions presented in skew coordinates Meromorphicity of ζB;γ

The meromorphic extension ζ of ζ0

B satisfies

S∗ = g−1 and has poles on

• λ

− 1

2n

2n eπij/n : 0 ≤ j ≤ 2n − 1, n ≥ 1

• with the

natural boundary |s| = g−1. htop(B) = log g → |s| = g−1 = e−htop(B).

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks

Square lattice Ising model with finite range interactions

Ising model: (Ruelle’s formulation)

(i) External field H (ii) The coupling constant J in the horizontal direction (iii) The coupling constant J ′ in the vertical direction (iv) Each site (i, j) of the square lattice Z2 has a spin ui,j with two possible values, +1 or −1. (v) The state space Σ(B) is given by B ⊆ {+1, −1}Z2×2.

ui,j ui−1,j ui+1,j ui,j+1 ui,j+1 ui,j−1 J ′ J ′ J J H

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks

Given L =

• n

l k

• Z2 ∈ L2, the Hamiltonian (energy) E(U) for

n l k

• periodic U = (ui,j) is defined by

E(U) = −J

• 0≤i≤k−1

0≤j≤n−1

ui,jui+1,j − J ′

• 0≤i≤k−1

0≤j≤n−1

ui,jui,j+1 − H

• 0≤i≤k−1

0≤j≤n−1

ui,j. The partition function for the n l k

• periodic and B-admissible

states is defined by ZB;L = ZB n l k

• ≡
• U∈fixL(Σ(B))

exp [−E(U)/kBT ] , where kB is Boltzmann’s constant and T is the temperature.

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks

[Lind] Thermodynamic zeta function for Z2 Ising model with finite range intersections is given by ζ0

B;Ising(s) ≡ exp L∈L2

ZB;L s[L] [L]

• (1)

= exp ∞

• n=1

∞

• k=1

n−1

• l=0

1 nk ZB n l k

• snk
• .

(2) For any n ≥ 1, define the n-th order thermodynamic zeta function ζIsing;B;n(s) as ζIsing;B;n(s) ≡ exp

• 1

n

∞

• k=1

n−1

• l=0

1 k ZB n l k

• snk
• ;

(3) The thermodynamic zeta function ζIsing;B(s) is given by ζIsing;B(s) ≡

∞

• n=1

ζIsing;B;n(s). (4)

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks

Theorem 3.1 Given any B ⊂ Σ2×2, ζIsing;B;n(s) = (det (I − snτIsing;n))−1 , (5) and ζIsing;B(s) =

∞

• n=1

(det (I − snτIsing;n))−1 . (6) Notably, the result also holds for all γ ∈ GL2(Z).

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks

Further remarks:

(I) Zeta functions for two-dimensional sofic shifts (W.G. Hu and S.S. Lin) (In preparation) (i) bounded-to-one (ii) finite-to-one 1-d Golden-Mean Shift 1-d Even Shift

1 a b b 1 Pn(E) = Pn(G) − (−1)n

ζG(s) =

1 (1−gs)(1−¯ gs)

ζE(s) =

1+s (1−gs)(1−¯ gs)

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks

(II) Natural boundary Lind’s conjecture: topological entropy h ⇒ radius of natural boundary =e−h Question: Σ(B) is strongly irreducible ⇒ Lind’s conjecture holds?

Ref: J.C. Ban, W.G. Hu, S.S. Lin and Y.H. Lin, Verification of mixing properties in two-dimensional shifts of finite type, submitted.

Wen-Guei Hu Two-dimensional zeta functions

Introduction Main results Ising model Further remarks

Thank you.

Wen-Guei Hu Two-dimensional zeta functions