Germs of analytic families of diffeomorphisms unfolding a parabolic point (I) Christiane Rousseau Work done with C. Christopher, P. Mardeˇ si´ c, R. Roussarie and L. Teyssier 1 Minicourse 1, Toulouse, November 2010
Structure of the mini-course ◮ Statement of the problem (first lecture) ◮ The preparation of the family (first lecture) ◮ Construction of a modulus of analytic classification in the codimension 1 case (second lecture) ◮ The realization problem in the codimension 1 case (third lecture) 2 Minicourse 1, Toulouse, November 2010
Statement of the problem We consider germs of generic k -parameter families f ǫ of diffeomorphisms unfolding a parabolic point of codimension k f 0 ( z ) = z + z k + 1 + o ( z k + 1 ) When are two such germs conjugate? 3 Statement of the problem Minicourse 1, Toulouse, November 2010
Conjugacy of two germs of families Two germs of families of diffeomorphisms f ǫ and ˜ f ˜ ǫ are conjugate it there exists r ,ρ > 0 and analytic functions H : D r × D ρ → C h : D ρ → C , such that ◮ h is a diffeomorphism and for each fixed ǫ , H ǫ = H ( · ,ǫ ) is a diffeomorphism; ◮ for all ǫ ∈ D ρ and for all z ∈ D r , then f h ( ǫ ) = H ǫ ◦ f ǫ ◦ ( H ǫ ) − 1 ˜ 4 Statement of the problem Minicourse 1, Toulouse, November 2010
The choice of D r D r is chosen so that the behaviour of f 0 on the boundary is as in 5 Statement of the problem Minicourse 1, Toulouse, November 2010
The choice of D r D r is chosen so that the behaviour of f 0 on the boundary is as in D ρ is chosen sufficiently small so that f ǫ has the same behaviour near the boundary. In particular, all fixed points of f ǫ remain inside the disk. 6 Statement of the problem Minicourse 1, Toulouse, November 2010
A natural strategy: the use of normal forms A germ of generic k -parameter family f ǫ unfolding a parabolic point of codimension k is formally conjugate to the time-1 map of a vector field P ǫ ( z ) ∂ v ǫ = 1 + a ( ǫ ) z k ∂ z where P ǫ ( z ) = z k + 1 + ǫ k − 1 z k − 1 + ··· + ǫ 1 z + ǫ 0 7 Statement of the problem Minicourse 1, Toulouse, November 2010
A natural strategy: the use of normal forms A germ of generic k -parameter family f ǫ unfolding a parabolic point of codimension k is formally conjugate to the time-1 map of a vector field P ǫ ( z ) ∂ v ǫ = 1 + a ( ǫ ) z k ∂ z where P ǫ ( z ) = z k + 1 + ǫ k − 1 z k − 1 + ··· + ǫ 1 z + ǫ 0 Problem: the change to normal form diverges. What does it mean? 8 Statement of the problem Minicourse 1, Toulouse, November 2010
Can we exploit the formal normal form despite its divergence? Let us look at the case k = 1: z 2 − ǫ ∂ v ǫ = 1 + a ( ǫ ) z ∂ z 9 Statement of the problem Minicourse 1, Toulouse, November 2010
Can we exploit the formal normal form despite its divergence? Let us look at the case k = 1: z 2 − ǫ ∂ v ǫ = 1 + a ( ǫ ) z ∂ z Two singular points ±√ ǫ with eigenvalues ± 2 √ ǫ 1 ± a ( ǫ ) √ ǫ µ ± = 10 Statement of the problem Minicourse 1, Toulouse, November 2010
The parameter is an analytic invariant of the vector field! Indeed, we have 1 + 1 = a ( ǫ ) µ + µ − 1 − 1 = 1 √ ǫ µ + µ − 11 Statement of the problem Minicourse 1, Toulouse, November 2010
Hence, can we hope to bring the system to a “prenormal” form in which the parameter is invariant? 12 Statement of the problem Minicourse 1, Toulouse, November 2010
Hence, can we hope to bring the system to a “prenormal” form in which the parameter is invariant? Yes! This is the preparation part. 13 Statement of the problem Minicourse 1, Toulouse, November 2010
Hence, can we hope to bring the system to a “prenormal” form in which the parameter is invariant? Yes! This is the preparation part. Advantage: a conjugacy between prepared families must preserve the canonical parameters. 14 Statement of the problem Minicourse 1, Toulouse, November 2010
Theorem We consider a diffeomorphism with a parabolic point of codimension k: f 0 ( z ) = z + z k + 1 + o ( z k + 1 ) For any generic k-parameter unfolding f η , there exists an analytic change of coordinate and parameter ( z ,η ) � → ( Z ,ǫ ) in a neighborhood of the origin transforming the family into the prepared form F ǫ ( Z ) = Z + P ǫ ( Z )( 1 + Q ǫ ( Z )+ P ǫ ( Z ) K ( Z ,ǫ )) 15 The preparation of the family Minicourse 1, Toulouse, November 2010
Theorem We consider a diffeomorphism with a parabolic point of codimension k: f 0 ( z ) = z + z k + 1 + o ( z k + 1 ) For any generic k-parameter unfolding f η , there exists an analytic change of coordinate and parameter ( z ,η ) � → ( Z ,ǫ ) in a neighborhood of the origin transforming the family into the prepared form F ǫ ( Z ) = Z + P ǫ ( Z )( 1 + Q ǫ ( Z )+ P ǫ ( Z ) K ( Z ,ǫ )) such that, if Z 1 ,... Z k + 1 are the fixed points, then � P ′ � ǫ ( Z j ) F ′ ǫ ( Z j ) = exp 1 + a ( ǫ ) Z k j 16 The preparation of the family Minicourse 1, Toulouse, November 2010
This determines almost uniquely the parameters! The only freedom will be inherited from a rotation of order k in Z τ k = 1 Z � → τ Z ; which yields the corresponding change on ǫ : ( ǫ k − 1 ,ǫ k − 2 ...,ǫ 0 ) � → ( τ 2 − k ǫ k − 1 ,τ 1 − k ǫ k − 2 ,...,τǫ 0 ) 17 The preparation of the family Minicourse 1, Toulouse, November 2010
Proof of the theorem We consider a diffeomorphism with a parabolic point of codimension k : f 0 ( z ) = z + z k + 1 + o ( z k + 1 ) A k -parameter unfolding can be written in the form f η ( z ) = z + p η ( z ) g η ( z ) , with g η ( z ) = 1 + O ( η, z ) . 18 The preparation of the family Minicourse 1, Toulouse, November 2010
Using the Weierstrass division theorem on the rest allows to write f η in the form f η ( z ) = z + p η ( z )( 1 + q η ( z )+ p η ( z ) h η ( z )) 19 The preparation of the family Minicourse 1, Toulouse, November 2010
Using the Weierstrass division theorem on the rest allows to write f η in the form f η ( z ) = z + p η ( z )( 1 + q η ( z )+ p η ( z ) h η ( z )) with p η ( z ) = z k + 1 + ν k − 1 ( η ) z k − 1 + ν 1 ( η ) z + ν 0 ( η ) 20 The preparation of the family Minicourse 1, Toulouse, November 2010
Using the Weierstrass division theorem on the rest allows to write f η in the form f η ( z ) = z + p η ( z )( 1 + q η ( z )+ p η ( z ) h η ( z )) with p η ( z ) = z k + 1 + ν k − 1 ( η ) z k − 1 + ν 1 ( η ) z + ν 0 ( η ) and q η ( z ) = c 0 ( η )+ c 1 ( η ) z + ··· + c k ( η ) z k . 21 The preparation of the family Minicourse 1, Toulouse, November 2010
Using the Weierstrass division theorem on the rest allows to write f η in the form f η ( z ) = z + p η ( z )( 1 + q η ( z )+ p η ( z ) h η ( z )) with p η ( z ) = z k + 1 + ν k − 1 ( η ) z k − 1 + ν 1 ( η ) z + ν 0 ( η ) and q η ( z ) = c 0 ( η )+ c 1 ( η ) z + ··· + c k ( η ) z k . Genericity condition: the Jacobian ∂ν ∂η is invertible. 22 The preparation of the family Minicourse 1, Toulouse, November 2010
Since f η ( z ) = z + p η ( z )( 1 + q η ( z )+ p η ( z ) h η ( z )) the fixed points z j of f η are the zeroes of p η . 23 The preparation of the family Minicourse 1, Toulouse, November 2010
The strategy The formal normal form is the time one map of a vector field P ǫ ( Z ) ∂ V ǫ = 1 + a ( ǫ ) Z k ∂ Z 24 The preparation of the family Minicourse 1, Toulouse, November 2010
The strategy The formal normal form is the time one map of a vector field P ǫ ( Z ) ∂ V ǫ = 1 + a ( ǫ ) Z k ∂ Z Hence the the fixed points of f η must be sent to the singular points Z j of V ǫ . 25 The preparation of the family Minicourse 1, Toulouse, November 2010
The strategy The formal normal form is the time one map of a vector field P ǫ ( Z ) ∂ V ǫ = 1 + a ( ǫ ) Z k ∂ Z Hence the the fixed points of f η must be sent to the singular points Z j of V ǫ . Moreover we need have f ′ η ( z j ) = exp ( V ′ ǫ ( Z j )) 26 The preparation of the family Minicourse 1, Toulouse, November 2010
How do we find the formal invariant a ( ǫ ) ? Let λ j = f ′ η ( z j ) We have that � 1 / ln ( λ j ) = a ( ǫ ) . 27 The preparation of the family Minicourse 1, Toulouse, November 2010
How do we find the formal invariant a ( ǫ ) ? Let λ j = f ′ η ( z j ) We have that � 1 / ln ( λ j ) = a ( ǫ ) . There exists a polynomial r η ( z ) of degree ≤ k such that at the points z j we have f ′ = p ′ � � ln η ( z j ) η ( z j )( 1 + r η ( z j )) . 28 The preparation of the family Minicourse 1, Toulouse, November 2010
How do we find the formal invariant a ( ǫ ) ? Let λ j = f ′ η ( z j ) We have that � 1 / ln ( λ j ) = a ( ǫ ) . There exists a polynomial r η ( z ) of degree ≤ k such that at the points z j we have f ′ = p ′ � � ln η ( z j ) η ( z j )( 1 + r η ( z j )) . (Such a polynomial is found by the Lagrange interpolation formula for distinct z j . The limit exists when two fixed points coallesce (codimension 1 case). We can fill in for the other values of η by Hartogs’s Theorem.) 29 The preparation of the family Minicourse 1, Toulouse, November 2010
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