Germs of analytic families of diffeomorphisms unfolding a parabolic point (III) Christiane Rousseau Work done with C. Christopher, P. Mardeˇ si´ c, R. Roussarie and L. Teyssier 1 Minicourse 3, 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 3, Toulouse, November 2010
The classification theorem Theorem. [MRR] Two germs of generic families unfolding a codimension 1 parabolic point are analytically conjugate if and only if they have same formal invariant a ( ǫ ) and same modulus �� � � Ψ 0 ǫ ,Ψ ∞ / ∼ ^ ǫ ^ ǫ ∈ V δ ^ 3 Minicourse 3, Toulouse, November 2010
The realization problem Which a ( ǫ ) and modulus �� � Ψ 0 � / ∼ are realizable? ǫ ,Ψ ∞ ^ ^ ǫ ^ ǫ ∈ V δ 4 Minicourse 3, Toulouse, November 2010
The strategy Ψ 0 � � 1. Any a ( ǫ ) and ǫ ,Ψ ∞ can be realized as the ^ ^ ǫ modulus of a diffeomorphism f ^ ǫ . This is the local realization . 5 The strategy Minicourse 3, Toulouse, November 2010
The strategy Ψ 0 � � 1. Any a ( ǫ ) and ǫ ,Ψ ∞ can be realized as the ^ ǫ ^ modulus of a diffeomorphism f ^ ǫ . This is the local realization . Ψ 0 � � 2. If a ( ǫ ) is analytic and ǫ ,Ψ ∞ depend ^ ǫ ^ analytically on ^ ǫ , then the realization f ^ ǫ can be made depending analytically on ^ ǫ ∈ V δ with uniform limit for ^ ǫ = 0. 6 The strategy Minicourse 3, Toulouse, November 2010
3. On the auto-intersection of V δ we let � ǫ = ^ ǫ ǫ e 2 π i ǫ = ^ ˜ A necessary condition for the realization by a uniform family is that f ǫ and f ˜ ǫ be conjugate. 7 The strategy Minicourse 3, Toulouse, November 2010
3. On the auto-intersection of V δ we let � ǫ = ^ ǫ ǫ e 2 π i ǫ = ^ ˜ A necessary condition for the realization by a uniform family is that f ǫ and f ˜ ǫ be conjugate. 4. This necessary condition, called the compatibility condition , is also sufficient and allows to “correct” f ^ ǫ to a uniform family. This is the global realization . 8 The strategy Minicourse 3, Toulouse, November 2010
The local realization for a fixed ^ ǫ The technique is standard: we realize on an abstract 1-dimensional complex manifold, which we recognize to be holomorphically equivalent to an open set of C . 9 The local realization Minicourse 3, Toulouse, November 2010
The local realization for a fixed ^ ǫ The technique is standard: we realize on an abstract 1-dimensional complex manifold, which we recognize to be holomorphically equivalent to an open set of C . Indeed, we consider the two sectors U ± ǫ , each endowed ^ U + with the model diffeomorphism U + U − 0 8 f ± ǫ , i.e. the time-1 map of the U − vector field 8 U + z 2 − ǫ ∂ 0 U − U + 8 v ǫ = U − 0 1 + a ( ǫ ) z ∂ z 10 The local realization Minicourse 3, Toulouse, November 2010
Ξ Ξ − − Ξ Ξ Ξ Ξ − − Ξ Ξ The gluing on U + ǫ ∩ U − ^ ǫ ^ This gluing must be compatible with f ± ǫ on the three parts of the intersection, U 0 ǫ and U C ǫ , U ∞ ǫ . ^ ^ ^ 11 The local realization Minicourse 3, Toulouse, November 2010
The gluing on U + ǫ ∩ U − ^ ^ ǫ This gluing must be compatible with f ± ǫ on the three parts of the intersection, U 0 ǫ and U C ǫ , U ∞ ǫ . ^ ^ ^ In the time coordinate W of v ǫ this gluing is simply given by ~ Ξ Ξ 8 8 U + U − U + 8 0 Ψ 0 U 0 U − on ǫ ^ ǫ ^ Ξ 0 ~ 0 Ξ on U ∞ Ψ ∞ Ξ 8 ^ ǫ ǫ ^ Ξ 8 U + U C T ^ on ǫ 0 U − ^ ǫ U + 8 U − Ξ 0 Ξ 0 which commutes with T 1 . The map T ^ ǫ is a translation: it is the Lavaurs map . 12 The local realization Minicourse 3, Toulouse, November 2010
The time W of v ǫ √ � ǫ + a ( ǫ ) 2 ln ( z 2 − ǫ ) , ǫ ln z − ^ 1 ǫ ǫ � = 0 , ^ √ √ W = q − 1 ǫ ( z ) = 2 ^ z + ^ ^ − 1 z + a ( 0 ) ln ( z ) , ǫ = 0 . ^ Ψ ~ 8 − Ψ ^ + Ω 0 Ω 0 ε ~ Ξ Ξ 8 8 ~ ~ Ω + Ω − ^ U + ^ ε ε U − U + 0 8 ∼ α 0 U − Ψ 0 Ψ 0 ~ Ξ 0 ~ 0 ^ Ξ ε Ξ 8 Ξ 8 Ψ 8 ^ ε 8 Ψ U + ^ ε 0 U − U + 8 _ Ω + Ω − ^ Ω + U − ^ ε ε ^ _ ε Ω − ^ Ξ Ξ 0 0 ε α 0 Ψ 0 ^ ε Ψ 0 ^ ε 13 The local realization Minicourse 3, Toulouse, November 2010
Why T ^ ǫ is a translation? Ψ ~ 8 Ψ + − ^ Ω 0 Ω 0 ε ~ ~ Ω + In the time coordinate W , Ω − ^ ^ ε ε it is a diffeomorphism ∼ α 0 Ψ 0 ~ Ψ 0 ^ ε commuting with T 1 on a strip of width larger then 8 Ψ ^ 8 ε Ψ ^ ε 1 going from Im W = − ∞ _ Ω + Ω − ^ ^ Ω + ε ε ^ _ ε to Im W = + ∞ . Ω − ^ ε α 0 Ψ 0 ^ ε Ψ 0 ^ ε 14 The local realization Minicourse 3, Toulouse, November 2010
The gluing in z -coordinate In the z -coordinate, the gluing is ~ Ξ Ξ 8 8 U + simply given by U − U + 8 0 U − Ξ 0 ~ 0 Ξ ǫ ◦ q − 1 Ξ 0 ǫ ◦ Ψ 0 U 0 ǫ = q ^ on ^ ^ ǫ ^ ǫ ^ Ξ 8 Ξ 8 ǫ ◦ q − 1 Ξ ∞ ǫ = q ^ ǫ ◦ Ψ ∞ on U ∞ U + ^ ^ ǫ ^ ^ ǫ 0 U − U + 8 U C id on U − ǫ ^ Ξ Ξ 0 0 15 The local realization Minicourse 3, Toulouse, November 2010
Behavior of the gluing near the fixed points Ξ 0 , ∞ ( z ) = id + ξ 0 , ∞ ( z ) with ǫ ^ ^ ǫ √ A √ � � � < C (^ � � ξ 0 � | ǫ | ^ ǫ ( z ) ǫ ) � z + ǫ ^ � � ^ � √ A √ � � � < C (^ � � | ǫ | ^ � ξ ∞ ǫ ( z ) ǫ ) � z − ǫ ^ � � ^ � 16 The local realization Minicourse 3, Toulouse, November 2010
The compatibility condition For ^ in the auto- ǫ Ψ ~ 8 − Ψ ^ + intersection of V δ we Ω 0 Ω 0 ε have two descriptions ~ ~ Ω + Ω − ^ ^ ε ε ∼ of the modulus. A α 0 Ψ 0 Ψ 0 ~ ^ ε necessary condition Ψ 8 ^ ε 8 Ψ ^ ε for realizability to a _ Ω + Ω − ^ Ω + ^ ε ε ^ _ ε uniform family in ǫ Ω − ^ ε α 0 is that they encode Ψ 0 ^ ε Ψ 0 ^ conjugate dynamics. ε 17 The compatibility condition Minicourse 3, Toulouse, November 2010
Parameter values in the auto-intersection For these values, the fixed points are linearizable and there is an orbit from one point to the other. 18 The compatibility condition Minicourse 3, Toulouse, November 2010
Parameter values in the auto-intersection For these values, the fixed points are linearizable and there is an orbit from one point to the other. This allows for a third description of the modulus studied by Glutsyuk. 19 The compatibility condition Minicourse 3, Toulouse, November 2010
Parameter values in the auto-intersection For these values, the fixed points are linearizable and there is an orbit from one point to the other. This allows for a third description of the modulus studied by Glutsyuk. Indeed, we can bring the diffeomorphism to the model (normal form) in the neighborhood of each fixed point by means of maps ϕ ± . The two normalization domains intersect. 20 The compatibility condition Minicourse 3, Toulouse, November 2010
Parameter values in the auto-intersection For these values, the fixed points are linearizable and there is an orbit from one point to the other. This allows for a third description of the modulus studied by Glutsyuk. Indeed, we can bring the diffeomorphism to the model (normal form) in the neighborhood of each fixed point by means of maps ϕ ± . The two normalization domains intersect. The Glutsyuk modulus is given by the comparison of the two normalizations ϕ − ◦ ( ϕ + ) − 1 21 The compatibility condition Minicourse 3, Toulouse, November 2010
Parameter values in the auto-intersection For these values, the fixed points are linearizable and there is an orbit from one point to the other. This allows for a third description of the modulus studied by Glutsyuk. Indeed, we can bring the diffeomorphism to the model (normal form) in the neighborhood of each fixed point by means of maps ϕ ± . The two normalization domains intersect. The Glutsyuk modulus is given by the comparison of the two normalizations ϕ − ◦ ( ϕ + ) − 1 The Glutsyuk modulus is unique up to composition on the left and on the right by maps of the form v t ǫ . 22 The compatibility condition Minicourse 3, Toulouse, November 2010
Construction of the Fatou Glutsyuk coordinates As before we construct Fatou Glutsyuk coordinates, Φ l and Φ r , but we use lines parallel to the line of holes 8 Ψ + − 0 0 Ψ 0 + − ^ ^ G ε ε Ψ ε ^ 23 The compatibility condition Minicourse 3, Toulouse, November 2010
Construction of the Fatou Glutsyuk coordinates As before we construct Fatou Glutsyuk coordinates, Φ l and Φ r , but we use lines parallel to the line of holes 8 Ψ + − 0 0 Ψ 0 + − ^ ^ G ε ε Ψ ε ^ The Glutsyuk modulus is Ψ G = Φ r ◦ ( Φ l ) − 1 It is unique up to composition on the left and on the right with translations and satisfies T α r ◦ Ψ G = Ψ G ◦ T α l 24 The compatibility condition Minicourse 3, Toulouse, November 2010
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