Irreductibility in Holomorphic Dynamics Xavier Buff Universit de - PowerPoint PPT Presentation

Irreductibility in Holomorphic Dynamics Xavier Buff Université de Toulouse joint work with Adam Epstein and Sarah Koch X. Buff Irreductibility in HD

Special curves P 3 is the dynamical moduli space of cubic polynomials modulo affine conjugacy. M 2 is the dynamical moduli space of quadratic rational maps modulo conjugacy by Möbius transformations. S k , n ⊂ P 3 (resp. V k , n ⊂ M 2 ) is the curve of conjugacy classes of cubic polynomials (resp. quadratic rational maps) having a critical point preperiodic to a cycle of period n with preperiod k . X. Buff Irreductibility in HD

S 0 , 3 X. Buff Irreductibility in HD

V 0 , 2 X. Buff Irreductibility in HD

V 2 , 1 X. Buff Irreductibility in HD

Irreductibility Conjecture (Milnor) For all n ≥ 1 , the curve S 0 , n is irreducible. X. Buff Irreductibility in HD

Irreductibility Conjecture (Milnor) For all n ≥ 1 , the curve S 0 , n is irreducible. Theorem (Arfeux-Kiwi) For all n ≥ 1 , the curve S 0 , n is irreducible. X. Buff Irreductibility in HD

Irreductibility Conjecture (Milnor) For all n ≥ 1 , the curve S 0 , n is irreducible. Theorem (Arfeux-Kiwi) For all n ≥ 1 , the curve S 0 , n is irreducible. Theorem (B.-Epstein-Koch) For all k ≥ 0 , the curve S k , 1 is irreducible. Theorem (B.-Epstein-Koch) For all k ≥ 2 , the curve V k , 1 is irreducible. X. Buff Irreductibility in HD

Equation of S k , 1 F a , b ( z ) = z 3 − 3 a 2 z + 2 a 3 + b , ( a , b ) ∈ C 2 . P 3 is obtained by identifying ( a , b ) with ( − a , − b ) . X. Buff Irreductibility in HD

Equation of S k , 1 F a , b ( z ) = z 3 − 3 a 2 z + 2 a 3 + b , ( a , b ) ∈ C 2 . P 3 is obtained by identifying ( a , b ) with ( − a , − b ) . P k := F ◦ k a , b ( a ) : k − 3 a 2 P k + 2 a 3 + b . P k + 1 = P 3 P 0 = a , P 1 = b and F a , b ( z ) − F a , b ( w ) = ( z − w )( z 2 + zw + w 2 − 3 a 2 ) . Q k := P 2 k − 1 + P k − 1 P k + P 2 k − 3 a 2 . X. Buff Irreductibility in HD

Equation of S k , 1 F a , b ( z ) = z 3 − 3 a 2 z + 2 a 3 + b , ( a , b ) ∈ C 2 . P 3 is obtained by identifying ( a , b ) with ( − a , − b ) . P k := F ◦ k a , b ( a ) : k − 3 a 2 P k + 2 a 3 + b . P k + 1 = P 3 P 0 = a , P 1 = b and F a , b ( z ) − F a , b ( w ) = ( z − w )( z 2 + zw + w 2 − 3 a 2 ) . Q k := P 2 k − 1 + P k − 1 P k + P 2 k − 3 a 2 . ( b − a ) divides Q k and so, Q k = ( b − a ) R k . X. Buff Irreductibility in HD

Equation of S k , 1 F a , b ( z ) = z 3 − 3 a 2 z + 2 a 3 + b , ( a , b ) ∈ C 2 . P 3 is obtained by identifying ( a , b ) with ( − a , − b ) . P k := F ◦ k a , b ( a ) : k − 3 a 2 P k + 2 a 3 + b . P k + 1 = P 3 P 0 = a , P 1 = b and F a , b ( z ) − F a , b ( w ) = ( z − w )( z 2 + zw + w 2 − 3 a 2 ) . Q k := P 2 k − 1 + P k − 1 P k + P 2 k − 3 a 2 . ( b − a ) divides Q k and so, Q k = ( b − a ) R k . Proposition The polynomial R k is irreducible. X. Buff Irreductibility in HD

Equation of S k , 1 R 1 = 2 a + b R 2 = ( 2 a + b ) 2 ( b − a ) 3 − 3 b ( 2 a + b )( a − b ) + 3 ( a + b ) . R 3 = ( 2 a + b ) 6 ( b − a ) 11 + · · · + 3 ( a + b ) . X. Buff Irreductibility in HD

Behavior near the origin From now on, k ≥ 2. Lemma The homogeneous part of least degree of R k is 3 ( a + b ) . X. Buff Irreductibility in HD

Behavior near the origin From now on, k ≥ 2. Lemma The homogeneous part of least degree of R k is 3 ( a + b ) . Corollary The polynomial R k ∈ Z [ a , b ] is irreducible over C if and only if it is irreducible over Q . X. Buff Irreductibility in HD

Behavior near the origin From now on, k ≥ 2. Lemma The homogeneous part of least degree of R k is 3 ( a + b ) . Corollary The polynomial R k ∈ Z [ a , b ] is irreducible over C if and only if it is irreducible over Q . Proof: the curve { R k = 0 } contains a non singular point with rational coordinates. X. Buff Irreductibility in HD

Behavior near infinity Lemma The homogeneous part of highest degree of R k is ( b − a ) 4 · 3 k − 2 − 1 · ( 2 a + b ) 2 · 3 k − 2 . Corollary The curve { R k = 0 } intersects the line at infinity at two points: [ 1 : 1 : 0 ] with multiplicity 4 · 3 k − 2 − 1 , and [ 1 : − 2 : 0 ] with multiplicity 2 · 3 k − 2 . X. Buff Irreductibility in HD

Intersection with the line { a = 0 } f b ( z ) := F 0 , b ( z ) = z 3 + b , X. Buff Irreductibility in HD

Intersection with the line { a = 0 } f b ( z ) := F 0 , b ( z ) = z 3 + b , p k ( b ) := P k ( 0 , b ) , q k ( b ) := Q k ( 0 , b ) and r k ( b ) := R k ( 0 , b ) . p k + 1 = p 3 k + b , q k + 1 = p 2 k + p k p k + 1 + p 2 k + 1 . q k = br k = bs k . Proposition (Goksel) The polynomial s k ∈ Z [ b ] is irreductible over Q . Proof: Work in F 3 [ b ] . p k ≡ b 3 k − 1 + b 3 k − 2 + · · · + b 3 + b ( mod 3 ) . p k + 1 − p k ≡ b 3 k ( mod 3 ) . s k ≡ b 2 · 3 k − 1 − 2 ( mod 3 ) . X. Buff Irreductibility in HD

Intersection with the line { a = 0 } f b ( z ) := F 0 , b ( z ) = z 3 + b , p k ( b ) := P k ( 0 , b ) , q k ( b ) := Q k ( 0 , b ) and r k ( b ) := R k ( 0 , b ) . p k + 1 = p 3 k + b , q k + 1 = p 2 k + p k p k + 1 + p 2 k + 1 . q k = br k = bs k . Proposition (Goksel) The polynomial s k ∈ Z [ b ] is irreductible over Q . Proof: Work in F 3 [ b ] . p k ≡ b 3 k − 1 + b 3 k − 2 + · · · + b 3 + b ( mod 3 ) . p k + 1 − p k ≡ b 3 k ( mod 3 ) . s k ≡ b 2 · 3 k − 1 − 2 ( mod 3 ) . Since s k ( 0 ) = 3, apply the Eisenstein criterion. X. Buff Irreductibility in HD