Torus actions in the normalization problem Jasmin Raissy Dipartimento di Matematica "L. Tonelli" Università di Pisa KAWA 1, Workshop Albi, January 29–31, 2010 Jasmin Raissy (Università di Pisa) Torus actions and normalization KAWA 1, 30 January 2010 1 / 29
Normalization Problem Setting Given f : ( C n , p ) → ( C n , p ) a germ of biholomorphism fixing p , we are interested in the dynamics of f in a neighbourhood of p . Jasmin Raissy (Università di Pisa) Torus actions and normalization KAWA 1, 30 January 2010 2 / 29
Normalization Problem Setting Given f : ( C n , p ) → ( C n , p ) a germ of biholomorphism fixing p , we are interested in the dynamics of f in a neighbourhood of p . i.e., for any q “sufficiently close” to p , we want to study the asymptotical behavior of { f k ( q ) } k ≥ 1 , where f k = f ◦ · · · ◦ f . Jasmin Raissy (Università di Pisa) Torus actions and normalization KAWA 1, 30 January 2010 2 / 29
Normalization Problem Setting Let f : ( C n , O ) → ( C n , O ) be germ of biholomorphism fixing O . Jasmin Raissy (Università di Pisa) Torus actions and normalization KAWA 1, 30 January 2010 3 / 29
Normalization Problem Setting Let f : ( C n , O ) → ( C n , O ) be germ of biholomorphism fixing O . Locally, using multi-index notation, � f Q z Q , f ( z ) = Λ z + Q ∈ N n | Q |≥ 2 where n � z Q := z q 1 1 · · · z q n n , f Q ∈ C n , | Q | := q j , j = 1 Jasmin Raissy (Università di Pisa) Torus actions and normalization KAWA 1, 30 January 2010 3 / 29
Normalization Problem Setting Let f : ( C n , O ) → ( C n , O ) be germ of biholomorphism fixing O . Locally, using multi-index notation, � f Q z Q , f ( z ) = Λ z + Q ∈ N n | Q |≥ 2 where n � z Q := z q 1 1 · · · z q n n , f Q ∈ C n , | Q | := q j , j = 1 with Λ in Jordan normal form, i.e., Λ = Diag ( λ 1 , . . . , λ n ) + N N = nilpotent matrix and λ 1 , . . . , λ n ∈ C ∗ not necessarily distinct. Jasmin Raissy (Università di Pisa) Torus actions and normalization KAWA 1, 30 January 2010 3 / 29
Normalization Problem Setting Let f : ( C n , O ) → ( C n , O ) be germ of biholomorphism fixing O . Locally, using multi-index notation, � f Q z Q , f ( z ) = Λ z + Q ∈ N n | Q |≥ 2 where n � z Q := z q 1 1 · · · z q n n , f Q ∈ C n , | Q | := q j , j = 1 with Λ in Jordan normal form, i.e., Λ = Diag ( λ 1 , . . . , λ n ) + N N = nilpotent matrix and λ 1 , . . . , λ n ∈ C ∗ not necessarily distinct. Want to know whether ∃ ϕ : ( C n , O ) → ( C n , O ) , local holomorphic change of coordinates, d ϕ O = Id , s.t. ϕ − 1 ◦ f ◦ ϕ has a simple form. Jasmin Raissy (Università di Pisa) Torus actions and normalization KAWA 1, 30 January 2010 3 / 29
Normalization Problem Linearization Linearization problem simple = linear Jasmin Raissy (Università di Pisa) Torus actions and normalization KAWA 1, 30 January 2010 4 / 29
Normalization Problem Linearization Linearization problem simple = linear Idea: first to search for a formal solution of (1) f ◦ ϕ = ϕ ◦ Λ and then to study the convergence of ϕ . Jasmin Raissy (Università di Pisa) Torus actions and normalization KAWA 1, 30 January 2010 4 / 29
Normalization Problem Linearization Linearization problem simple = linear Idea: first to search for a formal solution of (1) f ◦ ϕ = ϕ ◦ Λ and then to study the convergence of ϕ . We have to recursively solve, for each coordinate j , ϕ j ( z ) = z j + � | Q |≥ 2 ϕ Q , j z Q . λ Q := λ q 1 1 · · · λ q n n ( λ Q − λ j ) ϕ Q , j = Polynomial ( f P , j , ϕ R , k , with P ≤ Q , R < Q ) (2) Jasmin Raissy (Università di Pisa) Torus actions and normalization KAWA 1, 30 January 2010 4 / 29
Normalization Problem Linearization Linearization problem simple = linear Idea: first to search for a formal solution of (1) f ◦ ϕ = ϕ ◦ Λ and then to study the convergence of ϕ . We have to recursively solve, for each coordinate j , ϕ j ( z ) = z j + � | Q |≥ 2 ϕ Q , j z Q . λ Q := λ q 1 1 · · · λ q n n ( λ Q − λ j ) ϕ Q , j = Polynomial ( f P , j , ϕ R , k , with P ≤ Q , R < Q ) (2) lexicographic order on N n Jasmin Raissy (Università di Pisa) Torus actions and normalization KAWA 1, 30 January 2010 4 / 29
Normalization Problem Resonances Definition A resonant multi-index for λ ∈ ( C ∗ ) n , rel. to j ∈ { 1 , . . . , n } is Q ∈ N n , with | Q | ≥ 2, s.t. λ Q − λ j = 0 . (3) Res j ( λ ) := { Q ∈ N n | | Q | ≥ 2 , λ Q − λ j = 0 } . Jasmin Raissy (Università di Pisa) Torus actions and normalization KAWA 1, 30 January 2010 5 / 29
Normalization Problem Resonances Definition A resonant multi-index for λ ∈ ( C ∗ ) n , rel. to j ∈ { 1 , . . . , n } is Q ∈ N n , with | Q | ≥ 2, s.t. λ Q − λ j = 0 . (3) Res j ( λ ) := { Q ∈ N n | | Q | ≥ 2 , λ Q − λ j = 0 } . Resonances = obstruction to formal linearization. Jasmin Raissy (Università di Pisa) Torus actions and normalization KAWA 1, 30 January 2010 5 / 29
Normalization Problem Poincaré-Dulac normal forms Theorem (Poincaré-Dulac, 1904) ϕ O = Id , s.t. ∀ f as above ∃ � ϕ formal change of coord., d � ϕ − 1 ◦ f ◦ � ] n where g ( O ) = O, dg O = df O and g ϕ = g ∈ C [ [ z 1 , . . . , z n ] � has only resonant monomials, � g Q , j z Q . g j ( z ) = λ j z j + ε j z j + 1 + | Q |≥ 2 λ Q = λ j Moreover, the resonant terms of � ϕ can be arbitrarily chosen, and that choice determines uniquely g res and the remaining terms of � ϕ . A germ of the form Λ + g res , with g res containing only resonant monomials is said in Poincaré-Dulac normal form. Jasmin Raissy (Università di Pisa) Torus actions and normalization KAWA 1, 30 January 2010 6 / 29
Normalization Problem Normalization Problem Given f , ∃ ? ϕ : ( C n , O ) → ( C n , O ) , holomorphic change of coordinates, d ϕ O = Id , s.t. ϕ − 1 ◦ f ◦ ϕ is in Poincaré-Dulac normal form? Jasmin Raissy (Università di Pisa) Torus actions and normalization KAWA 1, 30 January 2010 7 / 29
Normalization Problem Normalization Problem Given f , ∃ ? ϕ : ( C n , O ) → ( C n , O ) , holomorphic change of coordinates, d ϕ O = Id , s.t. ϕ − 1 ◦ f ◦ ϕ is in Poincaré-Dulac normal form? Problem Not uniqueness of the formal change of coordinates � ϕ given by Poincaré-Dulac theorem, and not having explicit expression for g res , make very difficult to give estimates for the convergence of � ϕ . Jasmin Raissy (Università di Pisa) Torus actions and normalization KAWA 1, 30 January 2010 7 / 29
Torus Actions Jasmin Raissy (Università di Pisa) Torus actions and normalization KAWA 1, 30 January 2010 8 / 29
Torus Actions The same problem can be stated for germs of holomorphic vector field near a singular point. Jasmin Raissy (Università di Pisa) Torus actions and normalization KAWA 1, 30 January 2010 8 / 29
Torus Actions The same problem can be stated for germs of holomorphic vector field near a singular point. In 2002, N.T. Zung found that to find a Poincaré-Dulac holomorphic normalization for a germ of holomorphic vector field is the same as to find (and linearize) a suitable torus action which preserves the vector field. Jasmin Raissy (Università di Pisa) Torus actions and normalization KAWA 1, 30 January 2010 8 / 29
Torus Actions The same problem can be stated for germs of holomorphic vector field near a singular point. In 2002, N.T. Zung found that to find a Poincaré-Dulac holomorphic normalization for a germ of holomorphic vector field is the same as to find (and linearize) a suitable torus action which preserves the vector field. Idea We look for symmetries in the normalization problem and how to exploit them Jasmin Raissy (Università di Pisa) Torus actions and normalization KAWA 1, 30 January 2010 8 / 29
Torus Actions Germs commuting with a torus action Theorem (–, 2009) f commutes with a holom. effective T r -action on ( C n , O ) , 1 ≤ r ≤ n, with weight matrix Θ ∈ M n × r ( Z ) � ∃ ϕ local holom. change of coord. s.t. ϕ − 1 ◦ f ◦ ϕ contains only Θ -resonant monomials. Jasmin Raissy (Università di Pisa) Torus actions and normalization KAWA 1, 30 January 2010 9 / 29
Torus Actions Germs commuting with a torus action Theorem (–, 2009) f commutes with a holom. effective T r -action on ( C n , O ) , 1 ≤ r ≤ n, with weight matrix Θ ∈ M n × r ( Z ) � ∃ ϕ local holom. change of coord. s.t. ϕ − 1 ◦ f ◦ ϕ contains only Θ -resonant monomials. f commutes with A : T r × ( C n , O ) → ( C n , O ) , A ( x , O ) = O , means f ( A ( x , z )) = A ( x , f ( z )) . Jasmin Raissy (Università di Pisa) Torus actions and normalization KAWA 1, 30 January 2010 9 / 29
Torus Actions Germs commuting with a torus action Theorem (–, 2009) f commutes with a holom. effective T r -action on ( C n , O ) , 1 ≤ r ≤ n, with weight matrix Θ ∈ M n × r ( Z ) � ∃ ϕ local holom. change of coord. s.t. ϕ − 1 ◦ f ◦ ϕ contains only Θ -resonant monomials. f commutes with A : T r × ( C n , O ) → ( C n , O ) , A ( x , O ) = O , means f ( A ( x , z )) = A ( x , f ( z )) . A lin is semi-simple and Sp ( A lin ( x , · )) = { exp ( 2 π i � r k = 1 x k θ k j ) } j = 1 ,..., n where Θ = ( θ k j ) ∈ M n × r ( Z ) is the weight matrix of A . Jasmin Raissy (Università di Pisa) Torus actions and normalization KAWA 1, 30 January 2010 9 / 29
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