Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography A Pair of Pants Topologically pair of pants X is a 3-punctured sphere; 2 ℓ 1 2 ℓ 2 It is a surface of constant negative curvature − 1 and cannot be embedded into R 3 by Efimov’s theorem; 2 ℓ 3 10 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography A Pair of Pants Topologically pair of pants X is a 3-punctured sphere; 2 ℓ 1 2 ℓ 2 It is a surface of constant negative curvature − 1 and cannot be embedded into R 3 by Efimov’s theorem; As a metric space, it is uniquelly defined by the lengths of the three boundary 2 ℓ 3 geodesics: X = X ( ℓ 1 , ℓ 2 , ℓ 3 ) ; 10 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography A Pair of Pants Topologically pair of pants X is a 3-punctured sphere; 2 ℓ 1 2 ℓ 2 It is a surface of constant negative curvature − 1 and cannot be embedded into R 3 by Efimov’s theorem; As a metric space, it is uniquelly defined by the lengths of the three boundary 2 ℓ 3 geodesics: X = X ( ℓ 1 , ℓ 2 , ℓ 3 ) ; It possess countably many closed geodesics { γ n } of lengths 0 < ℓ ( γ 1 ) < ℓ ( γ 2 ) < . . . < ℓ ( γ n ) . . . → ∞ 10 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography A Pair of Pants Topologically pair of pants X is a 3-punctured sphere; 2 ℓ 1 2 ℓ 2 It is a surface of constant negative curvature − 1 and cannot be embedded into R 3 by Efimov’s theorem; As a metric space, it is uniquelly defined by the lengths of the three boundary 2 ℓ 3 geodesics: X = X ( ℓ 1 , ℓ 2 , ℓ 3 ) ; It possess countably many closed geodesics { γ n } of lengths 0 < ℓ ( γ 1 ) < ℓ ( γ 2 ) < . . . < ℓ ( γ n ) . . . → ∞ Symmetric pair of pants means ℓ 1 = ℓ 2 = ℓ 3 =: b . 10 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography The Hyperbolic Action Cutting the pair of pants along the red R 3 geodesics, we obtain a pair of hexagons; b R 1 b b R 2 11 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography The Hyperbolic Action Cutting the pair of pants along the red R 3 geodesics, we obtain a pair of hexagons; b The hexangons can be immersed into H 2 as R 1 b right-angled hexagons; b R 2 11 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography The Hyperbolic Action Cutting the pair of pants along the red R 3 geodesics, we obtain a pair of hexagons; b The hexangons can be immersed into H 2 as R 1 b right-angled hexagons; b The Fuchsian group Γ = � R 1 , R 2 , R 3 � , R 2 generated by reflections with respect to the “cuts”, gives a pair of pants as a double cover of the factor space X ( b ) = H 2 / Γ; 11 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography The Hyperbolic Action Cutting the pair of pants along the red R 3 geodesics, we obtain a pair of hexagons; b The hexangons can be immersed into H 2 as R 1 b right-angled hexagons; b The Fuchsian group Γ = � R 1 , R 2 , R 3 � , R 2 generated by reflections with respect to the “cuts”, gives a pair of pants as a double cover of the factor space X ( b ) = H 2 / Γ; To any geodesic X corresponds a geodesic in H ; for any closed geodesic γ there exists R γ ∈ Γ preserving γ . 11 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography The Hyperbolic Action Cutting the pair of pants along the red R 3 geodesics, we obtain a pair of hexagons; b The hexangons can be immersed into H 2 as R 1 b right-angled hexagons; b The Fuchsian group Γ = � R 1 , R 2 , R 3 � , R 2 generated by reflections with respect to the “cuts”, gives a pair of pants as a double cover of the factor space X ( b ) = H 2 / Γ; To any geodesic X corresponds a geodesic in H ; for any closed geodesic γ there exists R γ ∈ Γ preserving γ . The action Γ � H 2 is hyperbolic. 11 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Basic Facts 1 In 1992, Guillop´ e established that in the case of geometrically finite hyperbolic surfaces of infinite area, the infinite product Z X converges for ℜ ( s ) sufficiently large and has a meromorphic extension to C . 2 Zeros of the Selberg zeta function correspond to the poles of the Ruelle zeta function given by ζ ( s ): = Z X ( s + 1) 1 − e − s ℓ ( γ ) � − 1 � � = Z X ( s ) γ = primitive closed geodesic 12 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Basic Facts 1 In 1992, Guillop´ e established that in the case of geometrically finite hyperbolic surfaces of infinite area, the infinite product Z X converges for ℜ ( s ) sufficiently large and has a meromorphic extension to C . 2 Zeros of the Selberg zeta function correspond to the poles of the Ruelle zeta function given by ζ ( s ): = Z X ( s + 1) 1 − e − s ℓ ( γ ) � − 1 � � = Z X ( s ) γ = primitive closed geodesic 3 There exists the largest real zero δ , which is equal to the Hausdorff dimension of the limit set of Γ (a subset of the unit circle). 12 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Basic Facts 1 In 1992, Guillop´ e established that in the case of geometrically finite hyperbolic surfaces of infinite area, the infinite product Z X converges for ℜ ( s ) sufficiently large and has a meromorphic extension to C . 2 Zeros of the Selberg zeta function correspond to the poles of the Ruelle zeta function given by ζ ( s ): = Z X ( s + 1) 1 − e − s ℓ ( γ ) � − 1 � � = Z X ( s ) γ = primitive closed geodesic 3 There exists the largest real zero δ , which is equal to the Hausdorff dimension of the limit set of Γ (a subset of the unit circle). 4 There is no other zeros with ℜ ( s ) = δ 12 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Basic Facts (continued) 5 δ is the growth rate of the number of primitive closed 1 geodesics δ = lim t →∞ t log # { γ : ℓ ( γ ) ≤ t } . 13 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Basic Facts (continued) 5 δ is the growth rate of the number of primitive closed 1 geodesics δ = lim t →∞ t log # { γ : ℓ ( γ ) ≤ t } . Moreover, # { γ : ℓ ( γ ) ≤ t } ∼ e δ t δ t . 13 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Basic Facts (continued) 5 δ is the growth rate of the number of primitive closed 1 geodesics δ = lim t →∞ t log # { γ : ℓ ( γ ) ≤ t } . Moreover, # { γ : ℓ ( γ ) ≤ t } ∼ e δ t δ t . 6 For a symmetric pair of pants δ = δ ( b ) ∼ 1 b (McMullen) 13 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Basic Facts (continued) 5 δ is the growth rate of the number of primitive closed 1 geodesics δ = lim t →∞ t log # { γ : ℓ ( γ ) ≤ t } . Moreover, # { γ : ℓ ( γ ) ≤ t } ∼ e δ t δ t . 6 For a symmetric pair of pants δ = δ ( b ) ∼ 1 b (McMullen) 7 There exists ε > 0 such that there is only finite number of zeros satisfying ℜ ( s ) > δ − ε (Jacobson–Naud) 13 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Basic Facts (continued) 5 δ is the growth rate of the number of primitive closed 1 geodesics δ = lim t →∞ t log # { γ : ℓ ( γ ) ≤ t } . Moreover, # { γ : ℓ ( γ ) ≤ t } ∼ e δ t δ t . 6 For a symmetric pair of pants δ = δ ( b ) ∼ 1 b (McMullen) 7 There exists ε > 0 such that there is only finite number of zeros satisfying ℜ ( s ) > δ − ε (Jacobson–Naud) 8 Complex zeros are related to the eigenvalues of the Laplacian operator acting on L 2 functions and are a subject of intensive research (Nonnenmacher, Patterson, Perry, Zworski . . . ). 13 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Basic Facts (continued) 5 δ is the growth rate of the number of primitive closed 1 geodesics δ = lim t →∞ t log # { γ : ℓ ( γ ) ≤ t } . Moreover, # { γ : ℓ ( γ ) ≤ t } ∼ e δ t δ t . 6 For a symmetric pair of pants δ = δ ( b ) ∼ 1 b (McMullen) 7 There exists ε > 0 such that there is only finite number of zeros satisfying ℜ ( s ) > δ − ε (Jacobson–Naud) 8 Complex zeros are related to the eigenvalues of the Laplacian operator acting on L 2 functions and are a subject of intensive research (Nonnenmacher, Patterson, Perry, Zworski . . . ). These are defined as the poles of the resolvent and are referred to as resonances of X . 13 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Closed Geodesics To every closed geodesic γ on X ( b ) cor- R 3 R 2 responds a cutting sequence of period 2 n R 1 R 2 R 3 R 1 · · · j 2 n − 1 j 2 n j 2 n +1 · · · , γ 12132 R 1 R 1 R 2 R 3 R 1 R 3 R 2 14 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Closed Geodesics To every closed geodesic γ on X ( b ) cor- R 3 R 2 responds a cutting sequence of period 2 n R 1 R 2 R 3 R 1 · · · j 2 n − 1 j 2 n j 2 n +1 · · · , γ 12132 where j k ∈ { 1 , 2 , 3 } , j k � = j k +1 for R 1 1 ≤ k ≤ 2 n and j 2 n � = j 1 . R 1 R 2 R 3 R 1 R 3 R 2 14 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Closed Geodesics To every closed geodesic γ on X ( b ) cor- R 3 R 2 responds a cutting sequence of period 2 n R 1 R 2 R 3 R 1 · · · j 2 n − 1 j 2 n j 2 n +1 · · · , γ 12132 where j k ∈ { 1 , 2 , 3 } , j k � = j k +1 for R 1 1 ≤ k ≤ 2 n and j 2 n � = j 1 . a periodic orbit of the subshift σ R 1 R 2 R 3 R 1 of finite type on the space of 3 symbols Σ = { 1 , 2 , 3 } Z with transition matrix R 3 R 2 0 1 1 1 0 1 1 1 0 14 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Transition Matrices Let’s fix n and define r n : Σ → R , r n (Σ) = ℓ ( γ [ σ [ n / 2] ,σ [ n / 2]+1 ]), where γ is chosen such that 15 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Transition Matrices Let’s fix n and define r n : Σ → R , r n (Σ) = ℓ ( γ [ σ [ n / 2] ,σ [ n / 2]+1 ]), where γ is chosen such that γ ′ { ℓ ( γ ′ ) | γ ′ intersects σ 1 , . . . σ n } ℓ ( γ ) = min 15 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Transition Matrices Let’s fix n and define r n : Σ → R , r n (Σ) = ℓ ( γ [ σ [ n / 2] ,σ [ n / 2]+1 ]), where γ is chosen such that γ ′ { ℓ ( γ ′ ) | γ ′ intersects σ 1 , . . . σ n } ℓ ( γ ) = min Let ξ 1 , . . . , ξ N be all subsequences of the sequences in Σ of the length n . 15 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Transition Matrices Let’s fix n and define r n : Σ → R , r n (Σ) = ℓ ( γ [ σ [ n / 2] ,σ [ n / 2]+1 ]), where γ is chosen such that γ ′ { ℓ ( γ ′ ) | γ ′ intersects σ 1 , . . . σ n } ℓ ( γ ) = min Let ξ 1 , . . . , ξ N be all subsequences of the sequences in Σ of the length n . We define an N × N transition matrix � k +1 = ξ j if ξ i 1 , k ; for k = 1 , . . . , n − 1 A n i , j = 0 , otherwise. 15 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Transition Matrices Let’s fix n and define r n : Σ → R , r n (Σ) = ℓ ( γ [ σ [ n / 2] ,σ [ n / 2]+1 ]), where γ is chosen such that γ ′ { ℓ ( γ ′ ) | γ ′ intersects σ 1 , . . . σ n } ℓ ( γ ) = min Let ξ 1 , . . . , ξ N be all subsequences of the sequences in Σ of the length n . We define an N × N transition matrix � k +1 = ξ j if ξ i 1 , k ; for k = 1 , . . . , n − 1 A n i , j = 0 , otherwise. and a complex matrix function A i , j ( s ) = exp( − sr n ( ξ )) · A n A : C → Mat ( N , N ) i , j , where ξ = ξ i 1 . . . ξ i n . . . ξ j n . 15 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Key Lemma Lemma 1 − e − s ℓ ( γ ) � 2 = lim � � � I N − A 2 ( s ) � n →∞ det ; γ = primitive closed geodesic where I N is the N × N identity matrix. 16 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Key Lemma Lemma 1 − e − s ℓ ( γ ) � 2 = lim � � � I N − A 2 ( s ) � n →∞ det ; γ = primitive closed geodesic where I N is the N × N identity matrix. Choosing n = 2 above we get r n = b det( Id − e − 2 sb A 2 ) = (1 − 4 e − 2 bs )(1 − e − 2 bs ) 2 16 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Key Lemma Lemma 1 − e − s ℓ ( γ ) � 2 = lim � � � I N − A 2 ( s ) � n →∞ det ; γ = primitive closed geodesic where I N is the N × N identity matrix. Choosing n = 2 above we get r n = b det( Id − e − 2 sb A 2 ) = (1 − 4 e − 2 bs )(1 − e − 2 bs ) 2 For a first approximation... The zero set belongs to a pair of straight lines 16 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Key Lemma Lemma 1 − e − s ℓ ( γ ) � 2 = lim � � � I N − A 2 ( s ) � n →∞ det ; γ = primitive closed geodesic where I N is the N × N identity matrix. Choosing n = 2 above we get r n = b det( Id − e − 2 sb A 2 ) = (1 − 4 e − 2 bs )(1 − e − 2 bs ) 2 For a first approximation... The zero set belongs to a pair of straight lines The distance between consequetive zeros is π b . 16 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Curves of Zeros — I Using n = 3 in the approximation of geodesics length r 3 ( ξ ) = b + c ( ξ ) e − b + O ( e − 2 b ) , 17 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Curves of Zeros — I Using n = 3 in the approximation of geodesics length r 3 ( ξ ) = b + c ( ξ ) e − b + O ( e − 2 b ) , we obtain a 6 × 6 matrix which determinant has the zero set on the curves � 1 � C 1 = 2 ln | 2 − 2 cos( t ) | + it | t ∈ R ; 17 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Curves of Zeros — I Using n = 3 in the approximation of geodesics length r 3 ( ξ ) = b + c ( ξ ) e − b + O ( e − 2 b ) , we obtain a 6 × 6 matrix which determinant has the zero set on the curves � 1 � C 1 = 2 ln | 2 − 2 cos( t ) | + it | t ∈ R ; � 1 � C 2 = 2 ln | 2 + cos(2 t ) | + it | t ∈ R ; 17 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Curves of Zeros — I Using n = 3 in the approximation of geodesics length r 3 ( ξ ) = b + c ( ξ ) e − b + O ( e − 2 b ) , we obtain a 6 × 6 matrix which determinant has the zero set on the curves � 1 � C 1 = 2 ln | 2 − 2 cos( t ) | + it | t ∈ R ; � 1 � C 2 = 2 ln | 2 + cos(2 t ) | + it | t ∈ R ; � 1 � � � � 1 − 1 2 e 2 it − 1 2 e it � � � 4 − 3 e 2 it C 3 = 2 ln � + it | t ∈ R ; � � 17 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Curves of Zeros — I Using n = 3 in the approximation of geodesics length r 3 ( ξ ) = b + c ( ξ ) e − b + O ( e − 2 b ) , we obtain a 6 × 6 matrix which determinant has the zero set on the curves � 1 � C 1 = 2 ln | 2 − 2 cos( t ) | + it | t ∈ R ; � 1 � C 2 = 2 ln | 2 + cos(2 t ) | + it | t ∈ R ; � 1 � � � � 1 − 1 2 e 2 it − 1 2 e it � � � 4 − 3 e 2 it C 3 = 2 ln � + it | t ∈ R ; � � � � � 1 � 1 − 1 2 e 2 it + 1 � 2 e it � � � C 4 = 2 ln 4 − 3 e 2 it � + it | t ∈ R . � � 17 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Curves of Zeros — II 1.5 0.9 ℑ ( s ) 0.85 1 0.8 0.75 0.5 0.7 0.65 0 0 0.5 1 0.3 0.35 0.4 0.45 ℜ ( s ) � σ b + ite b � Figure: The zero sets of ζ X (red) and the curves C k , � ln 2 2 , π � (black) for b = 6; and a zoomed neighbourhood of . 4 18 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Comments on Geometric Approximation 1 Increasing n we do not see a change in the zero set for ℑ ( z ) < e 3 b ; 19 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Comments on Geometric Approximation 1 Increasing n we do not see a change in the zero set for ℑ ( z ) < e 3 b ; 2 There is no good estimates on error term (or rate of convergence). 19 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Comments on Geometric Approximation 1 Increasing n we do not see a change in the zero set for ℑ ( z ) < e 3 b ; 2 There is no good estimates on error term (or rate of convergence). We need to estimate the approximation error. 19 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Transfer Operators Technique Given a hyperbolic action, we introduce: 20 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Transfer Operators Technique Given a hyperbolic action, we introduce: 1 A proper Banach space of analytic functions; 20 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Transfer Operators Technique Given a hyperbolic action, we introduce: 1 A proper Banach space of analytic functions; 2 A nuclear transfer operator acting on the Banach space; 20 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Transfer Operators Technique Given a hyperbolic action, we introduce: 1 A proper Banach space of analytic functions; 2 A nuclear transfer operator acting on the Banach space; 3 The determinant of the transfer operator, which is an analytic function; 20 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Transfer Operators Technique Given a hyperbolic action, we introduce: 1 A proper Banach space of analytic functions; 2 A nuclear transfer operator acting on the Banach space; 3 The determinant of the transfer operator, which is an analytic function; 4 Ruelle–Pollicott dynamical zeta function; 20 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Transfer Operators Technique Given a hyperbolic action, we introduce: 1 A proper Banach space of analytic functions; 2 A nuclear transfer operator acting on the Banach space; 3 The determinant of the transfer operator, which is an analytic function; 4 Ruelle–Pollicott dynamical zeta function; 5 The Ruelle zeta function turns to be an analytic function, which is closely related to the determinant (of the transfer operator); 20 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Transfer Operators Technique Given a hyperbolic action, we introduce: 1 A proper Banach space of analytic functions; 2 A nuclear transfer operator acting on the Banach space; 3 The determinant of the transfer operator, which is an analytic function; 4 Ruelle–Pollicott dynamical zeta function; 5 The Ruelle zeta function turns to be an analytic function, which is closely related to the determinant (of the transfer operator); 6 The zeta function can be computed very efficiently using periodic orbits data (of the hyperbolic system) and its zeros provide quontitative information about the system. 20 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography The Banach Space The space B of analytic functions on the union of disjoint disks ⊔ 3 k =1 U k , chosen so that R i ( U j ∪ U k ) ⊂ U i for any three distinct i , j , k ∈ { 1 , 2 , 3 } . 21 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography The Banach Space The space B of analytic functions on the union of disjoint disks ⊔ 3 k =1 U k , chosen so that R i ( U j ∪ U k ) ⊂ U i for any three distinct i , j , k ∈ { 1 , 2 , 3 } . U 1 β 1 β 3 b U 2 b b β 2 U 3 Figure: The domain of analytic functions forming the Banach space (in pale red). 21 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Transfer Operator We define a transfer operator L s on the space B by ( L s f ) | U 1 ( z 1 ) = | R ′ 1 ( z 2 ) | s f ( z 2 ) + | R ′ 1 ( z 3 ) | s f ( z 3 ) , where z 2 , z 3 are preimages of z 1 ∈ U 1 with respect to reflection with respect to the geodesic β 1 . 22 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Transfer Operator We define a transfer operator L s on the space B by ( L s f ) | U 1 ( z 1 ) = | R ′ 1 ( z 2 ) | s f ( z 2 ) + | R ′ 1 ( z 3 ) | s f ( z 3 ) , where z 2 , z 3 are preimages of z 1 ∈ U 1 with respect to reflection with respect to the geodesic β 1 . Lemma (Grothendieck–Ruelle) The operator L s is nuclear. 22 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Transfer Operator We define a transfer operator L s on the space B by ( L s f ) | U 1 ( z 1 ) = | R ′ 1 ( z 2 ) | s f ( z 2 ) + | R ′ 1 ( z 3 ) | s f ( z 3 ) , where z 2 , z 3 are preimages of z 1 ∈ U 1 with respect to reflection with respect to the geodesic β 1 . Lemma (Grothendieck–Ruelle) The operator L s is nuclear. We may write the determinant of the transfer operator as ∞ z n � � def � n Tr L n ζ ( z , s ) = exp − = det( I − z L s ) . s n =1 22 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Zeta Function Magic Lemma (Grothendieck–Ruelle) The trace of the transfer operator may be explicitly computed in terms of the closed geodesics. exp( − s ℓ ( γ )) � Tr L n s = 1 − exp( − ℓ ( γ )) | γ | = n 23 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Zeta Function Magic Lemma (Grothendieck–Ruelle) The trace of the transfer operator may be explicitly computed in terms of the closed geodesics. exp( − s ℓ ( γ )) � Tr L n s = 1 − exp( − ℓ ( γ )) | γ | = n Theorem (Ruelle) There exists a constant δ such that the determinant is an analytic function in both variables in a strip 0 < s < δ , and � ∞ 1 exp( − s ℓ ( γ )) � � � ζ (1 , s ) = ζ ( s ) = exp 1 − exp( − ℓ ( γ )) n n =1 | γ | = n 23 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography First Attempt on Location of Zeros Using Ruelle’s Theorem, ∞ N � � ζ ( s ) = a n ( s ) = lim a n ( s ) , N →∞ n =0 n =0 24 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography First Attempt on Location of Zeros Using Ruelle’s Theorem, ∞ N � � ζ ( s ) = a n ( s ) = lim a n ( s ) , N →∞ n =0 n =0 where a n are explicitely defined in terms of closed geodesics of the word length not more than | γ | ≤ 2 n , 24 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography First Attempt on Location of Zeros Using Ruelle’s Theorem, ∞ N � � ζ ( s ) = a n ( s ) = lim a n ( s ) , N →∞ n =0 n =0 where a n are explicitely defined in terms of closed geodesics of the word length not more than | γ | ≤ 2 n , and are analytic in s . 24 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography First Attempt on Location of Zeros Using Ruelle’s Theorem, ∞ N � � ζ ( s ) = a n ( s ) = lim a n ( s ) , N →∞ n =0 n =0 where a n are explicitely defined in terms of closed geodesics of the word length not more than | γ | ≤ 2 n , and are analytic in s . N Choosing truncation ζ N ( s ) = � a n ( s ), we can n =0 1 find the largest real zero = the width of the critical strip, 24 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography First Attempt on Location of Zeros Using Ruelle’s Theorem, ∞ N � � ζ ( s ) = a n ( s ) = lim a n ( s ) , N →∞ n =0 n =0 where a n are explicitely defined in terms of closed geodesics of the word length not more than | γ | ≤ 2 n , and are analytic in s . N Choosing truncation ζ N ( s ) = � a n ( s ), we can n =0 1 find the largest real zero = the width of the critical strip, 2 consider a dense lattice in the strip, 24 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography First Attempt on Location of Zeros Using Ruelle’s Theorem, ∞ N � � ζ ( s ) = a n ( s ) = lim a n ( s ) , N →∞ n =0 n =0 where a n are explicitely defined in terms of closed geodesics of the word length not more than | γ | ≤ 2 n , and are analytic in s . N Choosing truncation ζ N ( s ) = � a n ( s ), we can n =0 1 find the largest real zero = the width of the critical strip, 2 consider a dense lattice in the strip, 3 compute the residue over each square, 24 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography First Attempt on Location of Zeros Using Ruelle’s Theorem, ∞ N � � ζ ( s ) = a n ( s ) = lim a n ( s ) , N →∞ n =0 n =0 where a n are explicitely defined in terms of closed geodesics of the word length not more than | γ | ≤ 2 n , and are analytic in s . N Choosing truncation ζ N ( s ) = � a n ( s ), we can n =0 1 find the largest real zero = the width of the critical strip, 2 consider a dense lattice in the strip, 3 compute the residue over each square, 4 find a zero using Newton method starting from a point of the lattice. 24 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Numerical Output: Symmetric Pants 0.15 0.1 0.05 0 0 500 1000 1500 2000 2500 3000 3500 0.1156 0.11555 0.1155 0.11545 0.1154 0.11535 0.1153 0 5 10 15 20 25 30 35 40 Figure: Zeros of the zeta function associated to a symmetric pair of pants and a more careful look for b = 12, N = 14. 25 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Q&A 1 Is the zero set of ζ N close to the zero set of ζ ? 26 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Q&A 1 Is the zero set of ζ N close to the zero set of ζ ? 2 How can we prove this? 26 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Q&A 1 Is the zero set of ζ N close to the zero set of ζ ? 2 How can we prove this? 3 What are characteric properties of the set of zeros of ζ ? 26 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Q&A 1 Is the zero set of ζ N close to the zero set of ζ ? 2 How can we prove this? 3 What are characteric properties of the set of zeros of ζ ? 4 How can we explain them? 26 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Q&A 1 Is the zero set of ζ N close to the zero set of ζ ? → Yes! 2 How can we prove this? 3 What are characteric properties of the set of zeros of ζ ? 4 How can we explain them? 26 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Q&A 1 Is the zero set of ζ N close to the zero set of ζ ? → Yes! 2 How can we prove this? → Use transfer operators 3 What are characteric properties of the set of zeros of ζ ? 4 How can we explain them? 26 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Q&A 1 Is the zero set of ζ N close to the zero set of ζ ? → Yes! 2 How can we prove this? → Use transfer operators 3 What are characteric properties of the set of zeros of ζ ? Qualitative observations The vertical spacing of zeros is approximately π b . The pattern of zeros appears to lie on four distinct curves, which seem to have a common point at δ 2 + i π 2 e b The vertical apparent periodicity of the pattern of zeros is approximately π e b . 4 How can we explain them? 26 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Q&A 1 Is the zero set of ζ N close to the zero set of ζ ? → Yes! 2 How can we prove this? → Use transfer operators 3 What are characteric properties of the set of zeros of ζ ? Qualitative observations The vertical spacing of zeros is approximately π b . The pattern of zeros appears to lie on four distinct curves, which seem to have a common point at δ 2 + i π 2 e b The vertical apparent periodicity of the pattern of zeros is approximately π e b . 4 How can we explain them? → Study the very beginning of the geodesics length spectrum 26 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Another viewpoint: exponential sums The function ζ N ( s ) is a finite exponential sum n � ζ N ( s ) = α k exp( µ k s ) , j = k where the multipliers µ k are the lengths of closed geodesics with word length up to 2 N . 1 Zeros form a point-periodic set and belong to a finite strip, parallel to the imaginary axis 27 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Another viewpoint: exponential sums The function ζ N ( s ) is a finite exponential sum n � ζ N ( s ) = α k exp( µ k s ) , j = k where the multipliers µ k are the lengths of closed geodesics with word length up to 2 N . 1 Zeros form a point-periodic set and belong to a finite strip, parallel to the imaginary axis 2 Their imaginary parts satisfy relation π ℑ ( s k ) = + ϕ ( k ) , max µ k − min µ k for an almost periodic function ϕ . 27 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Main Approximation Result R ( T ) = { s ∈ C | 0 ≤ |ℜ ( s ) | ≤ δ and |ℑ ( s ) | ≤ T } . 28 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Main Approximation Result R ( T ) = { s ∈ C | 0 ≤ |ℜ ( s ) | ≤ δ and |ℑ ( s ) | ≤ T } . Theorem (M. Pollicott-P. V.) Let X be a symmetric pair of pants with boundary geodesics of the length ℓ ( γ 0 ) = 2 b. 28 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Main Approximation Result R ( T ) = { s ∈ C | 0 ≤ |ℜ ( s ) | ≤ δ and |ℑ ( s ) | ≤ T } . Theorem (M. Pollicott-P. V.) Let X be a symmetric pair of pants with boundary geodesics of the length ℓ ( γ 0 ) = 2 b. We may approximate ζ on the domain R ( T ) by a complex trigonometric polynomial ζ n so that sup R ( T ) | ζ − ζ n | ≤ η ( b , n , T ) , 28 / 31
Zeta Function Surfaces Geometric approximation Ergodic Tools Locating zeros Bibliography Main Approximation Result R ( T ) = { s ∈ C | 0 ≤ |ℜ ( s ) | ≤ δ and |ℑ ( s ) | ≤ T } . Theorem (M. Pollicott-P. V.) Let X be a symmetric pair of pants with boundary geodesics of the length ℓ ( γ 0 ) = 2 b. We may approximate ζ on the domain R ( T ) by a complex trigonometric polynomial ζ n so that sup R ( T ) | ζ − ζ n | ≤ η ( b , n , T ) , where T ( b ) = e k 0 b for some constant 1 < k 0 < 2 independent of b and n, such that 28 / 31
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