Orbit Dirichlet series and multiset permutations Angela Carnevale - PowerPoint PPT Presentation

Orbit Dirichlet series and multiset permutations Angela Carnevale Universität Bielefeld (joint work with C. Voll)

Orbit Dirichlet series Let X be a space and T : X → X a map. For n ∈ N { x , T ( x ) , T 2 ( x ) , . . . , T n ( x ) = x } = closed orbit of length n O T ( n ) = number of closed orbits of length n under T . The orbit Dirichlet series of T is the Dirichlet generating series ∞ � O T ( n ) n − s , d T ( s ) = n = 1 where s is a complex variable. ◮ If O T ( n ) = 1 for all n ❀ d T ( s ) = ζ ( s ) ◮ For r ∈ N , if O T r ( n ) = a n ( Z r ) = number of index n subgroups of Z r r − 1 � ❀ d T r ( s ) = ζ ( s − i ) i = 0

Products and periodic points n �→ O T ( n ) is multiplicative ❀ Orbit Dirichlet series satisfy an Euler product ∞ � � � O T ( p k ) p − ks d T ( s ) = d T , p ( s ) = p prime p prime k = 0 To find the orbit series of a product of maps, we first look at another sequence: � F T ( n ) = number of points of period n = d O T ( d ) d | n Möbius inversion gives � n � O T ( n ) = 1 � F T ( d ) µ n d d | n For any finite collection of maps T 1 , . . . , T r F T 1 × ... × T r ( n ) = F T 1 ( n ) · · · F T r ( n )

Orbit series of products of maps Goal. For a partition λ = ( λ 1 , . . . , λ m ) , compute � d T λ ( s ) = d T λ 1 ×···× T λ m ( s ) = d T λ , p ( s ) , p prime where O T λ i ( n ) = number of index n subgroups of Z λ i . For i = 1, . . . , m � λ i − 1 + k � � λ i + k � O T λ i ( p k ) = F T λ i ( p k ) = and k k p p � m � � λ i + k � ∞ � � � p − k − ks . ❀ d T λ ( s ) = k p p k = 0 i = 1

Multiset permutations m � Let λ = ( λ 1 , . . . , λ m ) be a partition of N = λ i . i = 1 S λ = set of all multiset permutations on { 1 , . . . , 1 , 2 , . . . , 2 , . . . , m , . . . , m } . � �� � � �� � � �� � λ 1 λ 2 λ m ◮ λ = ( 1, . . . , 1 ) = ( 1 m ) ❀ S m = permutations of the set { 1 , 2 , . . . , m } , ◮ λ = ( 2, 1 ) ❀ S λ = { 112 , 121 , 211 } For w ∈ S λ , w = w 1 . . . w N Des ( w ) = { i ∈ [ N − 1 ] | w i > w i + 1 } , descent set of w des ( w ) = | Des ( w ) | , number of descents � maj ( w ) = i , major index i ∈ Des ( w ) ◮ λ = ( 3, 2, 1 ) , w = 121231 ∈ S λ ❀ Des ( w ) = { 2, 5 } , des ( w ) = 2 and maj ( w ) = 7.

Euler-Mahonian distribution and orbit series Let λ = ( λ 1 , . . . , λ m ) be a partition of N = � λ i � x des ( w ) q maj ( w ) ∈ Z [ x , q ] C λ ( x , q ) = w ∈ S λ Theorem (MacMahon 1916) � m � � λ i + k � ∞ � � C λ ( x , q ) x k = . � N k i = 0 ( 1 − xq i ) q k = 0 i = 1 Theorem (C.-Voll 2016) � w ∈ S λ p (− 1 − s ) des ( w )+ maj ( w ) C λ ( p − 1 − s , p ) � � d T λ ( s ) = = . � N � N i = 1 ( 1 − p i − 1 − s ) i = 1 ( 1 − p i − 1 − s ) p prime p prime

Example: λ = ( 1 m ) S ( 1 m ) = S m = symmetric group on n letters, C ( 1 m ) ( x , q ) = Carlitz’s q -Eulerian polynomial, � � j ∈ Des ( w ) p j − 1 − s C ( 1 m ) ( p − 1 − s , p ) w ∈ S m d T ( 1 m ) , p ( s ) = i = 1 ( 1 − p i − 1 − s ) = � m � m i = 1 ( 1 − p i − 1 − s ) � m � � p i − 1 − s 1 � = 1 − p i − 1 − s . 1 − p m − 1 − s I I ⊆ [ m − 1 ] i ∈ I Is an istance of an " Igusa function " ❀ d T ( 1 m ) , p ( s ) | p → p − 1 = (− 1 ) m p m − 1 − s d T ( 1 m ) , p ( s ) .

Local functional equations λ = ( λ 1 , . . . , λ m ) is a rectangle if λ 1 = · · · = λ m . Theorem (C.-Voll) 1. Let p be a prime. For all r , m ∈ N , d T ( rm ) , p ( s ) | p → p − 1 = (− 1 ) rm p m ( r + 1 2 ) − r − rs d T ( rm ) , p ( s ) . 2. If λ is not a rectangle, then d T λ , p ( s ) does not satisfy a functional equation of the form d T λ , p ( s ) | p → p − 1 = ± p d 1 − d 2 s d T λ , p ( s ) for d 1 , d 2 ∈ N 0 . Proof 1. Symmetry of C ( r m ) ( x , q ) + involution on S ( r m ) 2. C λ ( x , 1 ) has constant term 1. It is monic if and only if λ is a rectangle.

Abscissae of convergence and growth Fact. For an Euler product � � � W ( p , p − s ) = c kj p k − js , c kj � = 0 p p ( k , j ) ∈ I � a + 1 � ◮ α = abscissa of convergence = max | ( a , b ) ∈ I b � a � ◮ Meromorphic continuation to { Re ( s ) > β } , β = max b | ( a , b ) ∈ I Theorem (C.-Voll) λ = ( λ 1 , . . . , λ m ) , N = � λ i i 1. α λ = abs. of conv. of d T λ ( s ) = N , meromorphic continuation to { Re ( s ) > N − 2 } 2. There exists a constant K λ ∈ R > 0 such that � O T λ ( ν ) ∼ K λ n N as n → ∞ . ν � n

Abscissae of convergence and growth In our case � � � � � c kj p k − js = C λ ( p − 1 − s , p ) = p maj ( w )−( 1 + s ) des ( w ) p p ( k , j ) ∈ I λ p w ∈ S λ � � maj ( w )− des ( w )+ 1 ◮ α = max | w ∈ S λ des ( w ) � � maj ( w )− des ( w ) ◮ β = max | w ∈ S λ des ( w ) Theorem (C.-Voll) λ = ( λ 1 , . . . , λ m ) , N = � λ i i 1. α λ = abs. of conv. of d T λ ( s ) = N , meromorphic continuation to { Re ( s ) > N − 2 } 2. There exists a constant K λ ∈ R > 0 such that � O T λ ( ν ) ∼ K λ n N as n → ∞ . ν � n

Abscissae of convergence and growth In our case � � � � � c kj p k − js = C λ ( p − 1 − s , p ) = p maj ( w )−( 1 + s ) des ( w ) p p p ( k , j ) ∈ I λ w ∈ S λ � � maj ( w )− des ( w )+ 1 ◮ α = max | w ∈ S λ = N − 1 des ( w ) � � maj ( w )− des ( w ) ◮ β = max | w ∈ S λ = N − 2 des ( w ) Proof λ = ( λ 1 , . . . , λ m ) , N = � λ i i � � 1 1. α λ = max N − 1, abscissa of convergence of = N . � N i = 1 ( 1 − p i − 1 − s ) 2. There exists a constant K λ ∈ R > 0 such that � O T λ ( ν ) ∼ K λ n N as n → ∞ ( Tauberian theorem ) . ν � n

Natural boundaries: an example λ = ( 2, 1, 1 ) ❀ m = 3, N = 4, β = 2 C λ ( X , Y ) = 1 + 2 Y + 3 XY + 2 X 2 Y + XY 2 + 2 X 2 Y 2 + X 3 Y 2 ( a , b ) ∈ I λ ⇔ ∃ w ∈ S λ | des ( w ) = b and maj ( w ) = a + b 2 2 3 2 • = I λ

Natural boundaries: an example λ = ( 2, 1, 1 ) ❀ m = 3, N = 4, β = 2 C λ ( X , Y ) = 1 + 2 Y + 3 XY + 2 X 2 Y + XY 2 + 2 X 2 Y 2 + X 3 Y 2 ( a , b ) ∈ I λ ⇔ ∃ w ∈ S λ | des ( w ) = b and maj ( w ) = a + b � C 1 λ ( X , Y ) = 1 + 2 X 2 Y , not "cyclotomic" ⇓ 2 Re ( s ) = β is a natural boundary β x 1 y = 2 3 2 • = I λ

Natural boundaries: an example λ = ( λ 1 , . . . , λ m ) ❀ N = � i λ i , β = N − 2 � X maj ( w )− des ( w ) Y des ( w ) C λ ( X , Y ) = w ∈ S λ ( a , b ) ∈ I λ ⇔ ∃ w ∈ S λ | des ( w ) = b and maj ( w ) = a + b � C 1 λ ( X , Y ) = 1 + ( m − 1 ) X β Y ❀ Re ( s ) = β is a natural boundary β x 1 y = • = I λ

Natural boundaries Theorem (C.-Voll) Assume that m > 2. Then the orbit Dirichlet series d T λ ( s ) has a natural boundary at { Re ( s ) = N − 2 } . For m = 2 and λ � = ( 1, 1 ) we conjecture that the same holds. We prove it subject to: Conjecture 1 For λ 1 > λ 2 � λ 1 �� λ 2 � λ 2 � (− 1 ) i C ( λ 1 , λ 2 ) (− 1, 1 ) = � = 0 i i i = 0 Conjecture 2 For λ = ( λ 1 , λ 1 ) , λ 1 ≡ 1 ( mod 2 ) C λ ( x , q ) = ( 1 + xq λ 1 ) C ′ λ ( x , q ) , where C ′ λ (− 1, 1 ) � = 0.