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 Introduction 1 Riemann zeta function 1-d dynamical zeta function Thermodynamic zeta function Dynamical zeta function for Z d -actions Main results 2 Rationality of ζ n Zeta functions presented in skew coordinates Meromorphicity of ζ B ; γ Ising model 3 Further remarks 4 Wen-Guei Hu Two-dimensional zeta functions
Introduction Riemann zeta function Main results 1-d dynamical zeta function Ising model Thermodynamic zeta function Dynamical zeta function for Z d -actions Further remarks Outline Introduction 1 Riemann zeta function 1-d dynamical zeta function Thermodynamic zeta function Dynamical zeta function for Z d -actions Main results 2 Rationality of ζ n Zeta functions presented in skew coordinates Meromorphicity of ζ B ; γ Ising model 3 Further remarks 4 Wen-Guei Hu Two-dimensional zeta functions
Introduction Riemann zeta function Main results 1-d dynamical zeta function Ising model Thermodynamic zeta function Dynamical zeta function for Z d -actions Further remarks Riemann zeta function (1) Riemann zeta function: ∞ � n − s . ζ ( s ) := (1) n =1 Euler product formula: � � 1 − p − s � − 1 . ζ ( s ) = (2) p : prime 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 Riemann zeta function Main results 1-d dynamical zeta function Ising model Thermodynamic zeta function Dynamical zeta function for Z d -actions Further remarks 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: x the number of primes up to x is ∼ log x . Wen-Guei Hu Two-dimensional zeta functions
Introduction Riemann zeta function Main results 1-d dynamical zeta function Ising model Thermodynamic zeta function Dynamical zeta function for Z d -actions Further remarks Outline Introduction 1 Riemann zeta function 1-d dynamical zeta function Thermodynamic zeta function Dynamical zeta function for Z d -actions Main results 2 Rationality of ζ n Zeta functions presented in skew coordinates Meromorphicity of ζ B ; γ Ising model 3 Further remarks 4 Wen-Guei Hu Two-dimensional zeta functions
Introduction Riemann zeta function Main results 1-d dynamical zeta function Ising model Thermodynamic zeta function Dynamical zeta function for Z d -actions Further remarks (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. Γ n ( φ ) : the number of fixed point of φ n . � ∞ � � Γ n ( φ ) s n ζ φ ( s ) := exp . (3) n n =1 zeta function is defined only if Γ n ( φ ) < ∞ for all n ≥ 1 . Wen-Guei Hu Two-dimensional zeta functions
Introduction Riemann zeta function Main results 1-d dynamical zeta function Ising model Thermodynamic zeta function Dynamical zeta function for Z d -actions Further remarks Product formula: � 1 − s | γ | � − 1 � ζ φ ( s ) = , (4) γ 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 ζ φ is a rational function . Wen-Guei Hu Two-dimensional zeta functions
Introduction Riemann zeta function Main results 1-d dynamical zeta function Ising model Thermodynamic zeta function Dynamical zeta function for Z d -actions Further remarks 1-dim shifts of finite type Color set S p = { 0 , 1 , · · · , p − 1 } , p ≥ 2 Basic set of admissible local patterns B ⊂ S Z 2 × 1 p Σ( B ) : the set of all global patterns on Z 1 that can be generated by B P n ( B ) , n ≥ 1 : the set of all n -periodic patterns that can be generated by B , i.e., ( x i ) ∞ i = −∞ ∈ Σ( B ) with x j = x j + n for all j ∈ Z . x 1 x n x 2 x 1 x 2 x n P n ( B ) = ♯ P n ( B ) Wen-Guei Hu Two-dimensional zeta functions
Introduction Riemann zeta function Main results 1-d dynamical zeta function Ising model Thermodynamic zeta function Dynamical zeta function for Z d -actions Further remarks Example: (Golden-Mean shift) Basic set of admissible local patterns: � � B G = , , 0 0 0 1 1 0 � 1 � 1 Transition matrix A G = 1 0 P 1 ( B G ) = { 0 ∞ } → P 1 ( B G ) = 1 = tr ( A G ) P 2 ( B G ) = { (00) ∞ , (01) ∞ , (10) ∞ } → P 2 ( B G ) = 3 = tr ( A 2 G ) . . . P n ( B G ) = tr ( A n ⇒ G ) Wen-Guei Hu Two-dimensional zeta functions
Introduction Riemann zeta function Main results 1-d dynamical zeta function Ising model Thermodynamic zeta function Dynamical zeta function for Z d -actions Further remarks √ √ Eigenvalues of A G : g = 1+ 5 g = 1 − 5 , ¯ 2 2 � ∞ s k − log(1 − s ) = k k =1 Then, � ∞ � � P n ( A G ) s k ζ A G ( s ) ≡ exp k k =1 � ∞ � � tr ( A k G ) s k = exp k k =1 � ∞ � � g k +¯ g k s k = exp k k =1 1 = (1 − gs )(1 − ¯ gs ) Wen-Guei Hu Two-dimensional zeta functions
Introduction Riemann zeta function Main results 1-d dynamical zeta function Ising model Thermodynamic zeta function Dynamical zeta function for Z d -actions Further remarks � 1 − s | γ | � − 1 � ζ A G ( s ) = , (5) γ where the product is taken over all periodic orbits γ of φ and | γ | denotes the number of points in γ . Example: → γ = { 0 ∞ } , | γ | = 1 → γ = { (01) ∞ , (10) ∞ } = { (01) ∞ , σ ((01) ∞ ) } , | γ | = 2 → γ = { (001) ∞ , (010) ∞ , (100) ∞ } = { (001) ∞ , σ ((001) ∞ ) , σ 2 ((001) ∞ ) } | γ | = 3 Then, � 1 − s | γ | � − 1 � 1 1 1 ζ A G ( s ) = = 1 − s · 1 − s 2 · 1 − s 3 · · · · (6) γ Wen-Guei Hu Two-dimensional zeta functions
Introduction Riemann zeta function Main results 1-d dynamical zeta function Ising model Thermodynamic zeta function Dynamical zeta function for Z d -actions Further remarks A : m × m transition matrix. � ∞ � � tr ( A k ) s k ζ A ( s ) := exp k k =1 = (det( I − sA )) − 1 � (1 − λs ) − χ ( λ ) , = (7) λ ∈ Σ( A ) χ ( λ ) : algebraic multiplicity. � ∞ k s k = log( I − sA ) − 1 A k (8) k =1 exp ( tr ( M )) = det (exp( M )) (9) Wen-Guei Hu Two-dimensional zeta functions
Introduction Riemann zeta function Main results 1-d dynamical zeta function Ising model Thermodynamic zeta function Dynamical zeta function for Z d -actions Further remarks Outline Introduction 1 Riemann zeta function 1-d dynamical zeta function Thermodynamic zeta function Dynamical zeta function for Z d -actions Main results 2 Rationality of ζ n Zeta functions presented in skew coordinates Meromorphicity of ζ B ; γ Ising model 3 Further remarks 4 Wen-Guei Hu Two-dimensional zeta functions
Introduction Riemann zeta function Main results 1-d dynamical zeta function Ising model Thermodynamic zeta function Dynamical zeta function for Z d -actions Further remarks (3) Ruelle: Thermodynamic zeta function (1978) [ D. Ruelle, Thermodynamic Formalism, Addison-Wesley, 1978.] � ∞ � � Z n ( θ, α ) s n ζ R ( s ) := exp , (10) n n =1 where � � n − 1 �� � � θ ( α k x ) Z n ( θ, α ) = exp (11) x ∈ Fix α n k =0 is a partition function with periodic boundary conditions. Wen-Guei Hu Two-dimensional zeta functions
Introduction Riemann zeta function Main results 1-d dynamical zeta function Ising model Thermodynamic zeta function Dynamical zeta function for Z d -actions Further remarks Outline Introduction 1 Riemann zeta function 1-d dynamical zeta function Thermodynamic zeta function Dynamical zeta function for Z d -actions Main results 2 Rationality of ζ n Zeta functions presented in skew coordinates Meromorphicity of ζ B ; γ Ising model 3 Further remarks 4 Wen-Guei Hu Two-dimensional zeta functions
Introduction Riemann zeta function Main results 1-d dynamical zeta function Ising model Thermodynamic zeta function Dynamical zeta function for Z d -actions Further remarks (4) J.C. Ban, S.S. Lin and Y.H. Lin (2005): Zeta functions for 2-d shifts of finite type. Basic lattice: Z 2 × 2 Set of symbols / colors: S p = { 0 , 1 , · · · , p − 1 } . In particular, S 2 = { 0 , 1 } = { , } Σ 2 × 2 ( p ) := S Z 2 × 2 : the set of all local patterns. p Basic admissible set B : B ⊂ Σ 2 × 2 ( p ) . Wen-Guei Hu Two-dimensional zeta functions
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