Holomorphic linearization of commuting germs of holomorphic maps

Transcription

1 Holomorphic linearization of commuting germs of holomorphic maps Jasmin Raissy Dipartimento di Matematica e Applicazioni Università degli Studi di Milano Bicocca AMS 2010 Fall Eastern Sectional Meeting Special Session on Several Complex Variables Syracuse, October 2 3, 2010 Jasmin Raissy (Università di Milano Bicocca) Linearization of commuting germs Syracuse, October 3, / 13

2 Linearization Problem Given f : (C n, p) (C n, p) a germ of biholomorphism, f (p) = p,? ϕ local holomorphic change of coordinates, s.t. ϕ 1 f ϕ = linear part Λ of f? Jasmin Raissy (Università di Milano Bicocca) Linearization of commuting germs Syracuse, October 3, / 13

3 Linearization Problem Given f : (C n, p) (C n, p) a germ of biholomorphism, f (p) = p,? ϕ local holomorphic change of coordinates, s.t. ϕ 1 f ϕ = linear part Λ of f? Classical Idea: first look for a solution of f ϕ = ϕ Λ in the setting of formal power series, and then check whether ϕ is convergent. Jasmin Raissy (Università di Milano Bicocca) Linearization of commuting germs Syracuse, October 3, / 13

4 Linearization Problem Given f : (C n, O) (C n, O) a germ of biholomorphism, f (O) = O, with linear part in Jordan normal form λ 1 ε 1 λ 2 Λ = λ 1,..., λ n C, ε j 0 λ j = λ j+1, ε n 1 λ n? ϕ local holomorphic change of coordinates, s.t. dϕ O = Id and ϕ 1 f ϕ = Λ? Jasmin Raissy (Università di Milano Bicocca) Linearization of commuting germs Syracuse, October 3, / 13

5 Linearization Problem Dimension 1 λ 1: f is always holomorphically linearizable Jasmin Raissy (Università di Milano Bicocca) Linearization of commuting germs Syracuse, October 3, / 13

6 Linearization Problem Dimension 1 λ 1: f is always holomorphically linearizable λ = e 2πip/q : f is holomorphically linearizable f q Id Jasmin Raissy (Università di Milano Bicocca) Linearization of commuting germs Syracuse, October 3, / 13

7 Linearization Problem Dimension 1 λ 1: f is always holomorphically linearizable λ = e 2πip/q : f is holomorphically linearizable f q Id λ = e 2πiθ, θ R \ Q: f is always formally linearizable Jasmin Raissy (Università di Milano Bicocca) Linearization of commuting germs Syracuse, October 3, / 13

8 Linearization Problem Dimension 1 λ 1: f is always holomorphically linearizable λ = e 2πip/q : f is holomorphically linearizable f q Id λ = e 2πiθ, θ R \ Q: f is always formally linearizable Brjuno condition for λ f is holormorphically linearizable Jasmin Raissy (Università di Milano Bicocca) Linearization of commuting germs Syracuse, October 3, / 13

9 Linearization Problem Dimension 1 λ 1: f is always holomorphically linearizable λ = e 2πip/q : f is holomorphically linearizable f q Id λ = e 2πiθ, θ R \ Q: f is always formally linearizable Brjuno condition for λ f is holormorphically linearizable Yoccoz: Brjuno condition for λ the quadratic polynomial λz + z 2 is holormorphically linearizable (and moreover λz + z 2 hol. lin. f (z) = λz + hol. lin) Jasmin Raissy (Università di Milano Bicocca) Linearization of commuting germs Syracuse, October 3, / 13

10 Linearization Problem Dimension n 2 Formal Obstruction A resonant multi-index for λ (C ) n, rel. to j {1,..., n} is Q N n, with Q = n h=1 q h 2, s.t. Λ Q λ j = 0 where Λ Q := λ q 1 1 λqn n. Res j (Λ) := {Q N n Q 2, Λ Q λ j = 0}. Jasmin Raissy (Università di Milano Bicocca) Linearization of commuting germs Syracuse, October 3, / 13

11 Linearization Problem Dimension n 2 Formal Obstruction A resonant multi-index for λ (C ) n, rel. to j {1,..., n} is Q N n, with Q = n h=1 q h 2, s.t. Λ Q λ j = 0 where Λ Q := λ q 1 1 λqn n. Res j (Λ) := {Q N n Q 2, Λ Q λ j = 0}. But there are formal, and holomorphic, linearization results also in presence of resonances Theorem (Rüssmann 2002, R. 2010) f formally linearizable + Brjuno reduced condition f holomorphically linearizable Jasmin Raissy (Università di Milano Bicocca) Linearization of commuting germs Syracuse, October 3, / 13

12 Simultaneous Linearization Simultaneous Linearization Problem Given h 2 germs of biholomorphisms f 1,..., f h of C n at the same fixed point? ϕ a local holomorphic change of coordinates conjugating f k to its linear part for each k = 1,..., h? Jasmin Raissy (Università di Milano Bicocca) Linearization of commuting germs Syracuse, October 3, / 13

13 Simultaneous Linearization Dimension 1 Arnol d: asked about the smoothness of a simultaneous linearization of such a system, Jasmin Raissy (Università di Milano Bicocca) Linearization of commuting germs Syracuse, October 3, / 13

14 Simultaneous Linearization Dimension 1 Arnol d: asked about the smoothness of a simultaneous linearization of such a system, and this was brilliantly answered by Herman (1979), and extended by Yoccoz (1984). Jasmin Raissy (Università di Milano Bicocca) Linearization of commuting germs Syracuse, October 3, / 13

15 Simultaneous Linearization Dimension 1 Arnol d: asked about the smoothness of a simultaneous linearization of such a system, and this was brilliantly answered by Herman (1979), and extended by Yoccoz (1984). Moser, 1990: raised the problem of smooth linearization of commuting circle diffeomorphisms in connection with the holonomy group of certain foliations of codimension 1; with the rapidly convergent Nash-Moser iteration scheme, he proved that if the rotation numbers of the diffeomorphisms satisfy a simultaneous Diophantine condition and if the diffeomorphisms are in some C -neighborhood of the corresponding rotations, then they are C -conjugated to rotations. Jasmin Raissy (Università di Milano Bicocca) Linearization of commuting germs Syracuse, October 3, / 13

16 Simultaneous Linearization Dimension 1 Arnol d: asked about the smoothness of a simultaneous linearization of such a system, and this was brilliantly answered by Herman (1979), and extended by Yoccoz (1984). Moser, 1990: raised the problem of smooth linearization of commuting circle diffeomorphisms in connection with the holonomy group of certain foliations of codimension 1; with the rapidly convergent Nash-Moser iteration scheme, he proved that if the rotation numbers of the diffeomorphisms satisfy a simultaneous Diophantine condition and if the diffeomorphisms are in some C -neighborhood of the corresponding rotations, then they are C -conjugated to rotations. Pérez-Marco, 1997: commuting systems of analytic or smooth circle diffeomorphisms are deeply related to commuting systems of germs of holomorphic functions. Jasmin Raissy (Università di Milano Bicocca) Linearization of commuting germs Syracuse, October 3, / 13

17 Simultaneous Linearization Dimension 1 Arnol d: asked about the smoothness of a simultaneous linearization of such a system, and this was brilliantly answered by Herman (1979), and extended by Yoccoz (1984). Moser, 1990: raised the problem of smooth linearization of commuting circle diffeomorphisms in connection with the holonomy group of certain foliations of codimension 1; with the rapidly convergent Nash-Moser iteration scheme, he proved that if the rotation numbers of the diffeomorphisms satisfy a simultaneous Diophantine condition and if the diffeomorphisms are in some C -neighborhood of the corresponding rotations, then they are C -conjugated to rotations. Pérez-Marco, 1997: commuting systems of analytic or smooth circle diffeomorphisms are deeply related to commuting systems of germs of holomorphic functions. Fayad and Khanin, 2009: a finite number of commuting smooth circle diffeomorphisms with simultaneously Diophantine rotation numbers are smoothly conjugated to rotations. Jasmin Raissy (Università di Milano Bicocca) Linearization of commuting germs Syracuse, October 3, / 13

18 Simultaneous Linearization Dimension n 2 Gramchev and Yoshino, 1999: simultaneous holomorphic linearization for pairwise commuting germs without simultaneous resonances, with diagonalizable linear parts, and under a simultaneous Diophantine condition (further studied by Yoshino, 2004) and a few more technical assumptions. Jasmin Raissy (Università di Milano Bicocca) Linearization of commuting germs Syracuse, October 3, / 13

19 Simultaneous Linearization Dimension n 2 Gramchev and Yoshino, 1999: simultaneous holomorphic linearization for pairwise commuting germs without simultaneous resonances, with diagonalizable linear parts, and under a simultaneous Diophantine condition (further studied by Yoshino, 2004) and a few more technical assumptions., 2009: h 2 germs f 1,..., f h of biholomorphisms of C n, fixing the origin, s.t. the linear part of f 1 is diagonalizable and f 1 commutes with f k for any k = 2,..., h, under certain arithmetic conditions on the eigenvalues of the linear part of f 1 and some restrictions on their resonances, are simultaneously holomorphically linearizable if and only if there exists a particular complex manifold invariant under f 1,..., f h. Jasmin Raissy (Università di Milano Bicocca) Linearization of commuting germs Syracuse, October 3, / 13

20 Three natural questions Q1: shape of simultaneous linearization Is it possible to say anything on the shape a (formal) simultaneous linearization can have? Jasmin Raissy (Università di Milano Bicocca) Linearization of commuting germs Syracuse, October 3, / 13

21 Three natural questions Q1: shape of simultaneous linearization Is it possible to say anything on the shape a (formal) simultaneous linearization can have? Q2: conditions on the eigenvalues Are there any conditions on the eigenvalues of the linear parts of h 2 germs of simultaneously formally linearizable biholomorphisms ensuring simultaneous holomorphic linearizability? Jasmin Raissy (Università di Milano Bicocca) Linearization of commuting germs Syracuse, October 3, / 13

22 Three natural questions Q1: shape of simultaneous linearization Is it possible to say anything on the shape a (formal) simultaneous linearization can have? Q2: conditions on the eigenvalues Are there any conditions on the eigenvalues of the linear parts of h 2 germs of simultaneously formally linearizable biholomorphisms ensuring simultaneous holomorphic linearizability? Q3: generalization of Moser s question Under which conditions on the eigenvalues of the linear parts of h 2 pairwise commuting germs of biholomorphisms can one assert the existence of a simultaneous holomorphic linearization of the given germs? In particular, is there a Brjuno-type condition sufficient for convergence? Jasmin Raissy (Università di Milano Bicocca) Linearization of commuting germs Syracuse, October 3, / 13

23 Q1: shape of simultaneous linearization Proposition (, 2010) Let f 1,..., f h be h 2 formally linearizable germs of biholomorphisms of (C n, O), with almost simultaneously Jordanizable linear parts. f 1,..., f h simultaneously formally linearizable =!ϕ formal simultaneous linearization s.t. ϕ Q,j = 0 Q, j : Q h k=1 Res j(λ k ). Jasmin Raissy (Università di Milano Bicocca) Linearization of commuting germs Syracuse, October 3, / 13

24 Q1: shape of simultaneous linearization Definition M 1,..., M h, h 2 complex n n matrices are almost simultaneously Jordanizable, if a linear change of coordinates A s.t. A 1 M 1 A,..., A 1 M h A are almost in simultaneous Jordan normal form, i.e., for k = 1,..., h we have λ k,1 A 1 ε k,1 λ k,2 M k A =......, ε k,j 0 λ k,j = λ k,j+1. (1) ε k,n 1 λ k,n M 1,..., M h are simultaneously Jordanizable if a linear change of coordinates A s.t. we have (1) with ε k,j {0, ε}. Jasmin Raissy (Università di Milano Bicocca) Linearization of commuting germs Syracuse, October 3, / 13

25 Q1: shape of simultaneous linearization Proposition (, 2010) Let f 1,..., f h be h 2 formally linearizable germs of biholomorphisms of (C n, O), with almost simultaneously Jordanizable linear parts. f 1,..., f h simultaneously formally linearizable =!ϕ formal simultaneous linearization s.t. ϕ Q,j = 0 Q, j : Q h k=1 Res j(λ k ). Condition ensuring formal simultaneous linearizability: Jasmin Raissy (Università di Milano Bicocca) Linearization of commuting germs Syracuse, October 3, / 13

26 Q1: shape of simultaneous linearization Proposition (, 2010) Let f 1,..., f h be h 2 formally linearizable germs of biholomorphisms of (C n, O), with almost simultaneously Jordanizable linear parts. f 1,..., f h simultaneously formally linearizable =!ϕ formal simultaneous linearization s.t. ϕ Q,j = 0 Q, j : Q h k=1 Res j(λ k ). Condition ensuring formal simultaneous linearizability: Jasmin Raissy (Università di Milano Bicocca) Linearization of commuting germs Syracuse, October 3, / 13

27 Q1: shape of simultaneous linearization Proposition (, 2010) Let f 1,..., f h be h 2 formally linearizable germs of biholomorphisms of (C n, O), with almost simultaneously Jordanizable linear parts. f 1,..., f h simultaneously formally linearizable =!ϕ formal simultaneous linearization s.t. ϕ Q,j = 0 Q, j : Q h k=1 Res j(λ k ). Condition ensuring formal simultaneous linearizability: Theorem (, 2010) Let f 1,..., f h be h 2 formally linearizable germs of biholomorphisms of (C n, O), with almost simultaneously Jordanizable linear parts. If f p f q = f q f p p, q = f 1,..., f h simultaneously formally linearizable. Jasmin Raissy (Università di Milano Bicocca) Linearization of commuting germs Syracuse, October 3, / 13

28 Q1: shape of simultaneous linearization Condition ensuring formal simultaneous linearizability: Theorem (, 2010) Let f 1,..., f h be h 2 formally linearizable germs of biholomorphisms of (C n, O), with almost simultaneously Jordanizable linear parts. If f p f q = f q f p p, q = f 1,..., f h simultaneously formally linearizable. The hypothesis on the pairwise commutation is indeed necessary: If Λ 1 and Λ 2 are two commuting matrices almost in simultaneous Jordan n.f. s.t. Res(Λ 1 ) and Res(Λ 2 ), but Res(Λ 1 ) Res(Λ 2 ) =, the unique formal transformation tangent to the identity and commuting with both Λ 1 and Λ 2 is the identity, so any non-linear germ f 3 with linear part in Jordan normal form and commuting with Λ 1 (i.e., containing only Λ 1 -resonant terms) but not with Λ 2 cannot be simultaneously linearizable with Λ 1 and Λ 2. Jasmin Raissy (Università di Milano Bicocca) Linearization of commuting germs Syracuse, October 3, / 13

29 Q2: conditions on the eigenvalues Theorem (, 2010) Let f 1,..., f h be h 2 simultaneously formally linearizable germs of biholomorphism of (C n, O) s.t. their linear parts Λ 1,..., Λ h are simultaneously diagonalizable. If f 1,..., f h satisfy the simultaneous Brjuno condition, then f 1,... f h are holomorphically simultaneously linearizable. Jasmin Raissy (Università di Milano Bicocca) Linearization of commuting germs Syracuse, October 3, / 13

30 Q2: conditions on the eigenvalues Definition Λ 1 = (λ 1,1,..., λ 1,n ),..., Λ h = (λ h,1,..., λ h,n ), h 2 n-tuples of complex, not nec. distinct, non-zero numbers, satisfy the simultaneous Brjuno condition if ν ν log 1 ω Λ1,...,Λ h (2 ν+1 ) < +, where m 2 with ω Λ1,...,Λ h (m) := min ε Q, 2 Q m Q h k=1 n j=1 Res j (Λ k ) ε Q = max min Λ Q 1 k h 1 j n k λ k,j. Q h k=1 n j=1 Res j (Λ k ) If Λ 1,..., Λ h are the sets of eigenvalues of the linear parts of f 1,..., f h, we say that f 1,..., f h satisfy the simultaneous Brjuno condition. Jasmin Raissy (Università di Milano Bicocca) Linearization of commuting germs Syracuse, October 3, / 13

31 Q2: conditions on the eigenvalues Theorem (, 2010) Let f 1,..., f h be h 2 simultaneously formally linearizable germs of biholomorphism of (C n, O) s.t. their linear parts Λ 1,..., Λ h are simultaneously diagonalizable. If f 1,..., f h satisfy the simultaneous Brjuno condition, then f 1,... f h are holomorphically simultaneously linearizable. Jasmin Raissy (Università di Milano Bicocca) Linearization of commuting germs Syracuse, October 3, / 13

32 Q3: generalization of Moser s question Using the previous result we can give a positive answer to Q3. Jasmin Raissy (Università di Milano Bicocca) Linearization of commuting germs Syracuse, October 3, / 13

33 Q3: generalization of Moser s question Using the previous result we can give a positive answer to Q3. Theorem (, 2010) Let f 1,..., f h be h 2 formally linearizable germs of biholomorphisms of (C n, O) with simultaneously diagonalizable linear parts, and satisfying the simultaneous Brjuno condition. Then f 1,..., f h are sim. hol. linearizable they all commute pairwise. Jasmin Raissy (Università di Milano Bicocca) Linearization of commuting germs Syracuse, October 3, / 13

34 Remark If Λ 1,..., Λ h do not satisfy the sim. Brjuno cond., then each of them does not satisfy the reduced Brjuno cond., i.e., ν ν log 1 ω Λk (2 ν+1 = +, k = 1,..., h, ) ω Λk (m) := min Λ Q 2 Q m k λ k,j. 1 j n Q Res j (Λ k ) Jasmin Raissy (Università di Milano Bicocca) Linearization of commuting germs Syracuse, October 3, / 13

35 Remark If Λ 1,..., Λ h do not satisfy the sim. Brjuno cond., then each of them does not satisfy the reduced Brjuno cond., i.e., ν ν log 1 ω Λk (2 ν+1 = +, k = 1,..., h, ) ω Λk (m) := min Λ Q 2 Q m k λ k,j. 1 j n Q Res j (Λ k ) In particular, if Λ 1,..., Λ h are simultaneously Cremer, i.e., lim sup m + 1 m log 1 ω Λ1,...,Λ h (m) = +, and hence they do not satisfy the sim. Brjuno cond., then at least one of them has to be Cremer, i.e., lim sup m + 1 m log 1 ω Λk (m) = +, and the other ones do not satisfy the reduced Brjuno condition. Jasmin Raissy (Università di Milano Bicocca) Linearization of commuting germs Syracuse, October 3, / 13

36 Remark If Λ 1,..., Λ h do not satisfy the sim. Brjuno cond., then each of them does not satisfy the reduced Brjuno cond., i.e., ν ν log 1 ω Λk (2 ν+1 = +, k = 1,..., h, ) ω Λk (m) := min Λ Q 2 Q m k λ k,j. 1 j n Q Res j (Λ k ) Furthermore (following Moser and Yoshino) it is possible to find Λ 1,..., Λ h satisfying the simultaneous Brjuno condition, with Λ k not satisfying the reduced Brjuno condition for any k = 1,..., h. Jasmin Raissy (Università di Milano Bicocca) Linearization of commuting germs Syracuse, October 3, / 13

37 Thanks! Jasmin Raissy (Università di Milano Bicocca) Linearization of commuting germs Syracuse, October 3, / 13

38 Almost simultaneous Jordanizability Deciding when two n n complex matrices are almost simultaneously Jordanizable is not as easy as when the two matrices are diagonalizable. Jasmin Raissy (Università di Milano Bicocca) Linearization of commuting germs Syracuse, October 3, / 13

39 Almost simultaneous Jordanizability Deciding when two n n complex matrices are almost simultaneously Jordanizable is not as easy as when the two matrices are diagonalizable. h 2 diagonalizable matrices are sim. diagonalizable they commute pairwise Jasmin Raissy (Università di Milano Bicocca) Linearization of commuting germs Syracuse, October 3, / 13

40 Almost simultaneous Jordanizability Deciding when two n n complex matrices are almost simultaneously Jordanizable is not as easy as when the two matrices are diagonalizable. h 2 diagonalizable matrices are sim. diagonalizable they commute pairwise if h 2 matrices commute pairwise they are simultaneously triangularizable Jasmin Raissy (Università di Milano Bicocca) Linearization of commuting germs Syracuse, October 3, / 13

41 Almost simultaneous Jordanizability Deciding when two n n complex matrices are almost simultaneously Jordanizable is not as easy as when the two matrices are diagonalizable. h 2 diagonalizable matrices are sim. diagonalizable they commute pairwise if h 2 matrices commute pairwise they are simultaneously triangularizable (but the converse is clearly false). Jasmin Raissy (Università di Milano Bicocca) Linearization of commuting germs Syracuse, October 3, / 13

42 Almost simultaneous Jordanizability Deciding when two n n complex matrices are almost simultaneously Jordanizable is not as easy as when the two matrices are diagonalizable. h 2 diagonalizable matrices are sim. diagonalizable they commute pairwise if h 2 matrices commute pairwise they are simultaneously triangularizable (but the converse is clearly false). if two matrices commute then this does not imply that they admit an almost simultaneous Jordan normal form, Jasmin Raissy (Università di Milano Bicocca) Linearization of commuting germs Syracuse, October 3, / 13

43 Almost simultaneous Jordanizability Deciding when two n n complex matrices are almost simultaneously Jordanizable is not as easy as when the two matrices are diagonalizable. h 2 diagonalizable matrices are sim. diagonalizable they commute pairwise if h 2 matrices commute pairwise they are simultaneously triangularizable (but the converse is clearly false). if two matrices commute then this does not imply that they admit an almost simultaneous Jordan normal form, and it is not true in general that two matrices almost in simultaneous Jordan normal form commute Jasmin Raissy (Università di Milano Bicocca) Linearization of commuting germs Syracuse, October 3, / 13

44 Examples The two matrices λ 0 0 µ 0 0 Λ = ε λ 0 M = δ µ λ β 0 µ λ, ε, µ, δ, β C commute, but they are not almost simultaneously Jordanizable. In fact A s.t. AMA 1 is almost in sim. Jordan n.f. with Λ we have µ 0 0 AM = ζ µ 0 A (M µi 3 ζ 0) 0 0 µ β ζ f + δ ζ e 0 0 A = d e f, and det A 0 βf + δe 0, h 0. g h δ β h Jasmin Raissy (Università di Milano Bicocca) Linearization of commuting germs Syracuse, October 3, / 13

45 Examples The two matrices λ 0 0 µ 0 0 Λ = ε λ 0 M = δ µ λ β 0 µ λ, ε, µ, δ, β C commute, but they are not almost simultaneously Jordanizable. In fact A s.t. AMA 1 is almost in sim. Jordan n.f. with Λ we have µ 0 0 AM = ζ µ 0 A (M µi 3 ζ 0) 0 0 µ β ζ ζ e 0 0 A = d e f, and det A 0 βf + δe 0, h 0. g h δ β h λ 0 0 AΛ = ξ λ 0 A = h = 0 = Λ, M not almost sim Jordanizable 0 0 λ Jasmin Raissy (Università di Milano Bicocca) Linearization of commuting germs Syracuse, October 3, / 13

46 Examples The two matrices λ 0 0 Λ = ε λ 0 0 ε λ µ 0 0 M = δ µ η λ, ε, µ, δ, η C are almost in simultaneous Jordan normal form, Jasmin Raissy (Università di Milano Bicocca) Linearization of commuting germs Syracuse, October 3, / 13

47 Examples The two matrices λ 0 0 Λ = ε λ 0 0 ε λ µ 0 0 M = δ µ η λ, ε, µ, δ, η C are almost in simultaneous Jordan normal form, but they do not commute. Jasmin Raissy (Università di Milano Bicocca) Linearization of commuting germs Syracuse, October 3, / 13