An overview of local dynamics in several complex variables

Transcription

1 An overview of local dynamics in several complex variables Liverpool, January 15, 2008 Marco Abate University of Pisa, Italy

2 Local dynamics

3 Local dynamics Dynamics about a fixed point

4 Local dynamics Dynamics about a fixed point Stable set (iterates do not escape)

5 Local dynamics Dynamics about a fixed point Stable set (iterates do not escape) Conjugacy helps

6 One complex variable f(z) = z+a r z r +..., f(0)=0, f 0 (0)=

7 One complex variable f(z) = z+a r z r +..., f(0)=0, f 0 (0)= : multiplier

8 One complex variable f(z) = z+a r z r +..., f(0)=0, f 0 (0)= : multiplier Hyperbolic: =0,1

9 One complex variable f(z) = z+a r z r +..., f(0)=0, f 0 (0)= : multiplier Hyperbolic: =0,1 Superattracting: =0

10 One complex variable f(z) = z+a r z r +..., f(0)=0, f 0 (0)= : multiplier Hyperbolic: =0,1 Superattracting: =0 Parabolic: =1, =e 2πiθ, θ Q

11 One complex variable f(z) = z+a r z r +..., f(0)=0, f 0 (0)= : multiplier Hyperbolic: =0,1 Superattracting: =0 Parabolic: =1, =e 2πiθ, θ Q Elliptic: =1, =e 2πiθ, θ Q

12 One complex variable Hyperbolic: =0,1

13 One complex variable Hyperbolic: =0,1 Königs (1884): f(z) = z+arz r +... is locally holomorphically conjugated to g(z) = z (it is holomorphically linearizable).

14 One complex variable Hyperbolic: =0,1 Königs (1884): f(z) = z+arz r +... is locally holomorphically conjugated to g(z) = z (it is holomorphically linearizable). 0< <1: attracting; >1: repelling

15 One complex variable Superattracting: =0

16 One complex variable Superattracting: =0 Böttcher (1904): f(z) = arz r +... is locally holomorphically conjugated to g(z) = z r

17 One complex variable Parabolic: =1, =e 2πiθ, θ Q

18 One complex variable Parabolic: =1, =e 2πiθ, θ Q Leau-Fatou flower theorem (1897, 1919): f(z) = z + arz r +... dynamical flower with r 1 petals

19 One complex variable Parabolic: =1, =e 2πiθ, θ Q Leau-Fatou flower theorem (1897, 1919): f(z) = z + arz r +... dynamical flower with r 1 petals Camacho (1978): f is locally topologically conjugated to g(z) = z + z r and formally conjugated to h(z) = z + z r + cz 2r 1

20 One complex variable Parabolic: =1, =e 2πiθ, θ Q Leau-Fatou flower theorem (1897, 1919): f(z) = z + arz r +... dynamical flower with r 1 petals Camacho (1978): f is locally topologically conjugated to g(z) = z + z r and formally conjugated to h(z) = z + z r + cz 2r 1 Écalle, Voronin (1981): (very complicated) holomorphic classification

21 One complex variable Elliptic: =1, =e 2πiθ, θ Q

22 One complex variable Elliptic: =1, =e 2πiθ, θ Q Siegel-Bryuno (1942, 1965): if B (full-measure subset of S 1 ) then all f(z) = z+arz r +... are holomorphically linearizable.

23 One complex variable Elliptic: =1, =e 2πiθ, θ Q Siegel-Bryuno (1942, 1965): if B (full-measure subset of S 1 ) then all f(z) = z+arz r +... are holomorphically linearizable. Cremer-Yoccoz (1927, 1988): if B (dense uncountable subset of S 1 ) then f(z) = z+z 2 is not holomorphically linearizable.

24 f(z) = Az+P r (z)+..., f(o)=o, dfo=a

25 f(z) = Az+P r (z)+..., f(o)=o, dfo=a sp(a): multipliers

26 f(z) = Az+P r (z)+..., f(o)=o, dfo=a sp(a): multipliers Hyperbolic: sp(a) C * \S 1

27 f(z) = Az+P r (z)+..., f(o)=o, dfo=a sp(a): multipliers Hyperbolic: sp(a) C * \S 1 Superattracting: A=O

28 f(z) = Az+P r (z)+..., f(o)=o, dfo=a sp(a): multipliers Hyperbolic: sp(a) C * \S 1 Superattracting: A=O Parabolic (tangent to identity): A = I

29 f(z) = Az+P r (z)+..., f(o)=o, dfo=a sp(a): multipliers Hyperbolic: sp(a) C * \S 1 Superattracting: A=O Parabolic (tangent to identity): A = I Elliptic...

30 f(z) = Az+P r (z)+..., f(o)=o, dfo=a sp(a): multipliers Hyperbolic: sp(a) C * \S 1 Superattracting: A=O Parabolic (tangent to identity): A = I Elliptic... Mixed cases...

31 Hyperbolic: sp(a) C * \S 1 Similarities

32 Hyperbolic: sp(a) C * \S 1 Similarities Perron-Hadamard Stable Manifold Theorem ( 1928): E s : sum of gen. eigenspaces of attracting eigenvalues E u : sum of gen. eigenspaces of repelling eigenvalues Then there are complex manifolds W s/u tangent to E s/u at the origin such that f k (z) O iff z W s and f k (z) O iff z W u

33 Hyperbolic: sp(a) C * \S 1 Similarities Perron-Hadamard Stable Manifold Theorem ( 1928): E s : sum of gen. eigenspaces of attracting eigenvalues E u : sum of gen. eigenspaces of repelling eigenvalues Then there are complex manifolds W s/u tangent to E s/u at the origin such that f k (z) O iff z W s and f k (z) O iff z W u Grobman-Hartman ( ): f is locally topologically conjugated to Az

34 Hyperbolic: sp(a) C * \S 1 Differences

35 Hyperbolic: sp(a) C * \S 1 Differences In general, f is not locally holomorphically conjugated to Az

36 Hyperbolic: sp(a) C * \S 1 Differences In general, f is not locally holomorphically conjugated to Az ( 1 ) k 1... ( n ) k n j

37 Hyperbolic: sp(a) C * \S 1 Differences In general, f is not locally holomorphically conjugated to Az ( 1 ) k 1... ( n ) k n j Resonances prevent formal linearization

38 Hyperbolic: sp(a) C * \S 1 Differences In general, f is not locally holomorphically conjugated to Az ( 1 ) k 1... ( n ) k n j Resonances prevent formal linearization Poincaré (1893): if f is attracting/repelling, then f is holomorphically linearizable iff it is formally linearizable

39 Hyperbolic: sp(a) C * \S 1 Differences In general, f is not locally holomorphically conjugated to Az ( 1 ) k 1... ( n ) k n j Resonances prevent formal linearization Poincaré (1893): if f is attracting/repelling, then f is holomorphically linearizable iff it is formally linearizable Poincaré-Dulac (1912): f is formally conjugated to a map with only resonant monomials

40 Hyperbolic: sp(a) C * \S 1 Differences In general, f is not locally holomorphically conjugated to Az ( 1 ) k 1... ( n ) k n j Resonances prevent formal linearization Poincaré (1893): if f is attracting/repelling, then f is holomorphically linearizable iff it is formally linearizable Poincaré-Dulac (1912): f is formally conjugated to a map with only resonant monomials Small divisors prevent holomorphic linearization

41 Aside (by popular demand): Fatou-Bieberbach domains

42 Aside (by popular demand): Fatou-Bieberbach domains Take f globally defined on C n.

43 Aside (by popular demand): Fatou-Bieberbach domains Take f globally defined on C n. Assume attracting, no resonances: we get a local holomorphic linearization φ.

44 Aside (by popular demand): Fatou-Bieberbach domains Take f globally defined on C n. Assume attracting, no resonances: we get a local holomorphic linearization φ. Extend φ to basin of attraction B= C n : the image is C n.

45 Aside (by popular demand): Fatou-Bieberbach domains Take f globally defined on C n. Assume attracting, no resonances: we get a local holomorphic linearization φ. Extend φ to basin of attraction B= C n : the image is C n. No critical points of f in B implies no critical points of φ in B.

46 Aside (by popular demand): Fatou-Bieberbach domains Take f globally defined on C n. Assume attracting, no resonances: we get a local holomorphic linearization φ. Extend φ to basin of attraction B= C n : the image is C n. No critical points of f in B implies no critical points of φ in B. n=1: φ covering of C, impossible.

47 Aside (by popular demand): Fatou-Bieberbach domains Take f globally defined on C n. Assume attracting, no resonances: we get a local holomorphic linearization φ. Extend φ to basin of attraction B= C n : the image is C n. No critical points of f in B implies no critical points of φ in B. n=1: φ covering of C, impossible. n>1 and f globally invertible: B is biholomorphic to C n but not C n!

48 Aside (by popular demand): Fatou-Bieberbach domains Take f globally defined on C n. Assume attracting, no resonances: we get a local holomorphic linearization φ. Extend φ to basin of attraction B= C n : the image is C n. No critical points of f in B implies no critical points of φ in B. n=1: φ covering of C, impossible. n>1 and f globally invertible: B is biholomorphic to C n but not C n! f(z,w)=(w/2 z 2,z)

49 Superattracting: A=O Differences

50 Superattracting: A=O Differences In general, f is not locally topologically conjugated to Pr(z)

51 Superattracting: A=O Differences In general, f is not locally topologically conjugated to Pr(z) Topological, holomorphic, formal classifications: wide open, as well as local dynamics. (Some results by Hubbard, Favre-Jonsson).

52 Parabolic (tangent to identity): f(z) = z+p r (z)+... Similarities

53 Parabolic (tangent to identity): f(z) = z+p r (z)+... Similarities Characteristic direction: P r (v)=cv

54 Parabolic (tangent to identity): f(z) = z+p r (z)+... Similarities Characteristic direction: P r (v)=cv Degenerate: c=0; Non-degenerate otherwise.

55 Parabolic (tangent to identity): f(z) = z+p r (z)+... Similarities Characteristic direction: P r (v)=cv Degenerate: c=0; Non-degenerate otherwise. (Orbit tangent to v implies characteristic.)

56 Parabolic (tangent to identity): f(z) = z+p r (z)+... Characteristic direction: P r (v)=cv Degenerate: c=0; Non-degenerate otherwise. (Orbit tangent to v implies characteristic.) Parabolic curve: 1-dimensional holomorphic f-invariant curve, attracted by O. Similarities

57 Parabolic (tangent to identity): f(z) = z+p r (z)+... Similarities Characteristic direction: P r (v)=cv Degenerate: c=0; Non-degenerate otherwise. (Orbit tangent to v implies characteristic.) Parabolic curve: 1-dimensional holomorphic f-invariant curve, attracted by O. It can be (asymptotically) tangent to v.

58 Parabolic (tangent to identity): f(z) = z+p r (z)+... Characteristic direction: P r (v)=cv Degenerate: c=0; Non-degenerate otherwise. (Orbit tangent to v implies characteristic.) Parabolic curve: 1-dimensional holomorphic f-invariant curve, attracted by O. It can be (asymptotically) tangent to v. Écalle-Hakim (1985, 1998): if v is a non-degenerate characteristic direction, then there is a Fatou flower for f tangent to v. Similarities

59 Parabolic (tangent to identity): f(z) = z+p r (z)+... Characteristic direction: P r (v)=cv Degenerate: c=0; Non-degenerate otherwise. (Orbit tangent to v implies characteristic.) Parabolic curve: 1-dimensional holomorphic f-invariant curve, attracted by O. It can be (asymptotically) tangent to v. Écalle-Hakim (1985, 1998): if v is a non-degenerate characteristic direction, then there is a Fatou flower for f tangent to v. Similarities A. (2001): if n=2 and O isolated fixed point, then there is a Fatou flower for f.

60 Parabolic (tangent to identity): f(z) = z+p r (z)+... Differences

61 Parabolic (tangent to identity): f(z) = z+p r (z)+... Differences In general, parabolic curves are 1-dimensional only.

62 Parabolic (tangent to identity): f(z) = z+p r (z)+... Differences In general, parabolic curves are 1-dimensional only. Topological, holomorphic, formal classifications: wide open, as well as local dynamics. (Some results by Écalle, A.-Tovena).

63 Elliptic...: f(z) = Az+P r (z)+... Similarities

64 Elliptic...: f(z) = Az+P r (z)+... Similarities ω(m)=inf{ ( 1 ) k 1... ( n ) k n j ; 2 k k n m}

65 Elliptic...: f(z) = Az+P r (z)+... Similarities ω(m)=inf{ ( 1 ) k 1... ( n ) k n j ; 2 k k n m} Brjuno (1971): if Σm2 m log ω(2 m 1 )<+1 (Brjuno condition) then f is holomorphically linearizable

66 Elliptic...: f(z) = Az+P r (z)+... Similarities ω(m)=inf{ ( 1 ) k 1... ( n ) k n j ; 2 k k n m} Brjuno (1971): if Σm2 m log ω(2 m 1 )<+1 (Brjuno condition) then f is holomorphically linearizable Pöschel (1986): partial linearization results

67 Elliptic...: f(z) = Az+P r (z)+... Differences

68 Elliptic...: f(z) = Az+P r (z)+... Differences Not known if Brjuno condition is necessary.

69 Elliptic...: f(z) = Az+P r (z)+... Differences Not known if Brjuno condition is necessary. Convergence of a Poincaré-Dulac normal form?

70 An overview of local dynamics in several complex variables Liverpool, January 15, 2008 Marco Abate University of Pisa, Italy