Puiseux series dynamics and leading monomials of escape regions Jan - PowerPoint PPT Presentation

Puiseux series dynamics and leading monomials of escape regions Jan Kiwi PUC, Chile Workshop on Moduli Spaces Associated to Dynamical Systems ICERM Providence, April 17, 2012

Parameter space Monic centered cubic polynomials with marked critical points:

Parameter space Monic centered cubic polynomials with marked critical points: f a , v : z �→ ( z − a ) 2 ( z + 2 a ) + v where ( a , v ) ∈ C 2 .

Parameter space Monic centered cubic polynomials with marked critical points: f a , v : z �→ ( z − a ) 2 ( z + 2 a ) + v where ( a , v ) ∈ C 2 . After identification of ( a , v ) with ( − a , − v ) one obtains the moduli space of cubic polynomials with marked critical points.

Parameter space Monic centered cubic polynomials with marked critical points: f a , v : z �→ ( z − a ) 2 ( z + 2 a ) + v where ( a , v ) ∈ C 2 . After identification of ( a , v ) with ( − a , − v ) one obtains the moduli space of cubic polynomials with marked critical points. Critical points of f a , v are ± a . Critical value: v = f a , v (+ a ). Co-critical value − 2 a , since v = f a , v (+ a ) = f a , v ( − 2 a ).

Periodic Curves For p ≥ 1 , the periodic curve S p consists of all ( a , v ) such that + a has period p under f a , v .

Periodic Curves For p ≥ 1 , the periodic curve S p consists of all ( a , v ) such that + a has period p under f a , v . S p ⊂ C 2 is a smooth affine algebraic curve of degree d p where n | p d n = 3 p − 1 . �

Periodic Curves For p ≥ 1 , the periodic curve S p consists of all ( a , v ) such that + a has period p under f a , v . S p ⊂ C 2 is a smooth affine algebraic curve of degree d p where n | p d n = 3 p − 1 . � What is its topology?

Periodic Curves For p ≥ 1 , the periodic curve S p consists of all ( a , v ) such that + a has period p under f a , v . S p ⊂ C 2 is a smooth affine algebraic curve of degree d p where n | p d n = 3 p − 1 . � What is its topology? Is S p irreducible?

Periodic Curves For p ≥ 1 , the periodic curve S p consists of all ( a , v ) such that + a has period p under f a , v . S p ⊂ C 2 is a smooth affine algebraic curve of degree d p where n | p d n = 3 p − 1 . � What is its topology? Is S p irreducible? Bonifant,-,Milnor: the Euler characteristic is S p is ( 2 − p ) d p .

Periodic Curves For p ≥ 1 , the periodic curve S p consists of all ( a , v ) such that + a has period p under f a , v . S p ⊂ C 2 is a smooth affine algebraic curve of degree d p where n | p d n = 3 p − 1 . � What is its topology? Is S p irreducible? Bonifant,-,Milnor: the Euler characteristic is S p is ( 2 − p ) d p . What is the Euler characteristic of the smooth compactification of S p ?

Periodic Curves For p ≥ 1 , the periodic curve S p consists of all ( a , v ) such that + a has period p under f a , v . S p ⊂ C 2 is a smooth affine algebraic curve of degree d p where n | p d n = 3 p − 1 . � What is its topology? Is S p irreducible? Bonifant,-,Milnor: the Euler characteristic is S p is ( 2 − p ) d p . What is the Euler characteristic of the smooth compactification of S p ? Requires to compute the number N p of “escape regions”. (De Marco and Schiff, De Marco-Pilgrim).

Escape Regions The connectedness locus � � f a , v ∈ S p | f n C ( S p ) = a , v ( − a ) �→ ∞ is compact.

Escape Regions The connectedness locus � � f a , v ∈ S p | f n C ( S p ) = a , v ( − a ) �→ ∞ is compact. The escape locus � f a , v ∈ S p | f n � E ( S p ) = a , v ( − a ) → ∞ is open and every connected component is unbounded.

Escape Regions The connectedness locus � � f a , v ∈ S p | f n C ( S p ) = a , v ( − a ) �→ ∞ is compact. The escape locus � f a , v ∈ S p | f n � E ( S p ) = a , v ( − a ) → ∞ is open and every connected component is unbounded. A escape region U is a connected component of E ( S p ): U is conformally isomorphic to punctured disk. The puncture is at ∞ U .

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Asymptotics of critical periodic orbit For f a , v ∈ U , let a 0 = + a �→ a 1 = v �→ · · · �→ a p − 1 �→ a 0 .

Asymptotics of critical periodic orbit For f a , v ∈ U , let a 0 = + a �→ a 1 = v �→ · · · �→ a p − 1 �→ a 0 . Dynamical space picture. After conjugacy, f a , v ( az ) / a , we have:

Asymptotics of critical periodic orbit For f a , v ∈ U , let a 0 = + a �→ a 1 = v �→ · · · �→ a p − 1 �→ a 0 . Dynamical space picture. After conjugacy, f a , v ( az ) / a , we have:  a + o ( a ) or   a j =   − 2 a + o ( a ) .  

Asymptotics of critical periodic orbit For f a , v ∈ U , let a 0 = + a �→ a 1 = v �→ · · · �→ a p − 1 �→ a 0 . Dynamical space picture. After conjugacy, f a , v ( az ) / a , we have:  a + o ( a ) or   a j =   − 2 a + o ( a ) .   Bonifant and Milnor: Do the leading terms of a j − a , for j = 1 , . . . , p − 1 determine U uniquely?

Leading monomials There exists µ ≥ 1 and a local coordinate ζ for U near ∞ such that a = 1 ζ µ .

Leading monomials There exists µ ≥ 1 and a local coordinate ζ for U near ∞ such that a = 1 ζ µ . Then, � k /µ a j − a � 1 c k ζ k = � � = holomorphic ( ζ ) = . c k a a k ≥ k 0 k ≥ k 0

Leading monomials There exists µ ≥ 1 and a local coordinate ζ for U near ∞ such that a = 1 ζ µ . Then, � k /µ a j − a � 1 c k ζ k = � � = holomorphic ( ζ ) = . c k a a k ≥ k 0 k ≥ k 0 Leading monomial � k 0 /µ � 1 m j = c k 0 . a

Theorem. The leading monomial vector � m = ( m 1 , . . . , m p − 1 , 0 ) determines the escape region uniquely.

Theorem. The leading monomial vector � m = ( m 1 , . . . , m p − 1 , 0 ) determines the escape region uniquely. Corollary. (Bonifant,-,Milnor) Assume that the leading monomial vector of U is ( c 1 a − k 1 /µ , . . . , c p − 1 a − k p − 1 /µ ) . Then, for all j , a j ∈ Q ( c 1 , . . . , c p − 1 )(( a − 1 /µ )) .

Equations For an escape region U , � k /µ � 1 � v = a 1 = (+ a or − 2 a ) + a · c k a k ≥ k 0 is a solution of f p a , v (+ a ) = + a (*) in some extension of Q (( 1 / a )).

Field Put t instead of 1 / a to get Q (( t )) and study solutions of (*) in the algebraic closure of Q (( t )): Q a ≪ t ≫ = ∪ Q a (( t 1 / m )) .

Field Put t instead of 1 / a to get Q (( t )) and study solutions of (*) in the algebraic closure of Q (( t )): Q a ≪ t ≫ = ∪ Q a (( t 1 / m )) . Q a ≪ t ≫ is endowed with c k t k / m | = e − ord 0 ( z ) = e − k 0 / m . � | z = k ≥ k 0

Field Put t instead of 1 / a to get Q (( t )) and study solutions of (*) in the algebraic closure of Q (( t )): Q a ≪ t ≫ = ∪ Q a (( t 1 / m )) . Q a ≪ t ≫ is endowed with c k t k / m | = e − ord 0 ( z ) = e − k 0 / m . � | z = k ≥ k 0 Complete it to get L .

Non-Archimedean problem For ν ∈ L , consider ψ ν ( z ) = t − 2 ( z − 1 )( z + 2 ) + ν ∈ L [ z ] .

Non-Archimedean problem For ν ∈ L , consider ψ ν ( z ) = t − 2 ( z − 1 )( z + 2 ) + ν ∈ L [ z ] . ν is periodic parameter if ω + = + 1 is periodic under ψ ν .

Non-Archimedean problem For ν ∈ L , consider ψ ν ( z ) = t − 2 ( z − 1 )( z + 2 ) + ν ∈ L [ z ] . ν is periodic parameter if ω + = + 1 is periodic under ψ ν . In this case: ω + = + 1 �→ ω + 1 = ν �→ · · · �→ ω + p − 1 �→ ω + .

Non-Archimedean problem For ν ∈ L , consider ψ ν ( z ) = t − 2 ( z − 1 )( z + 2 ) + ν ∈ L [ z ] . ν is periodic parameter if ω + = + 1 is periodic under ψ ν . In this case: ω + = + 1 �→ ω + 1 = ν �→ · · · �→ ω + p − 1 �→ ω + . It follows, + 1 + c · t k / m + h . o . t .   ω +  j =  − 2 + d · t ℓ/ n + h . o . t .   

Non-Archimedean problem For ν ∈ L , consider ψ ν ( z ) = t − 2 ( z − 1 )( z + 2 ) + ν ∈ L [ z ] . ν is periodic parameter if ω + = + 1 is periodic under ψ ν . In this case: ω + = + 1 �→ ω + 1 = ν �→ · · · �→ ω + p − 1 �→ ω + . It follows, + 1 + c · t k / m + h . o . t .   ω +  j =  − 2 + d · t ℓ/ n + h . o . t .    The corresponding leading monomial are  c · t k / m  m ( ω + j − ω + ) =    − 3   respectively.

Theorem. The leading monomial vector � m = ( m 1 , . . . , m p − 1 , 0 ) determines the periodic parameter ν uniquely.

Dynamical Space The filled Julia set of ψ ν is K ( ψ ν ) = { z ∈ L | ψ n ν ( z ) �→ ∞} .

Dynamical Space The filled Julia set of ψ ν is K ( ψ ν ) = { z ∈ L | ψ n ν ( z ) �→ ∞} . | ν | ≤ 1 = ⇒ ψ n ν ( z ) → ∞ when | z | > 1 .

Dynamical Space The filled Julia set of ψ ν is K ( ψ ν ) = { z ∈ L | ψ n ν ( z ) �→ ∞} . | ν | ≤ 1 = ⇒ ψ n ν ( z ) → ∞ when | z | > 1 . Level 0 dynamical ball D 0 = {| z | ≤ 1 } .