Escaping to infinity

Transcription

1 Escaping to infinity Gwyneth Stallard The Open University Women in Mathematics, INI Cambridge April 2013

2 Undergraduate King s College, Cambridge

3 Undergraduate King s College, Cambridge Met future husband. Particularly enjoyed analysis - especially complex analysis.

4 PhD Imperial College, London

5 PhD Imperial College, London Studied the iterates of analytic functions of the complex plane.

6 PhD Imperial College, London Studied the iterates of analytic functions of the complex plane. Married in 1990.

7 Hanging on Southampton

8 Hanging on Southampton Progressed from hourly teaching to 0.5 lectureship to full-time 1 year lectureship

9 Hanging on Southampton Progressed from hourly teaching to 0.5 lectureship to full-time 1 year lectureship Carried on writing papers on dimensions of Julia sets.

10 Getting on the ladder Open University Progressed from temporary research fellowship to permanent lectureship in 1995.

11 Getting on the ladder Open University Progressed from temporary research fellowship to permanent lectureship in Carried on writing papers on dimensions of Julia sets.

12 Getting on the ladder Open University Progressed from temporary research fellowship to permanent lectureship in Carried on writing papers on dimensions of Julia sets. Tim born in 1998, Abigail in months maternity leave and switched to working 0.5.

13 Complex dynamics f : C C is analytic f n is the nth iterate of f

14 Complex dynamics f : C C is analytic f n is the nth iterate of f Definition The escaping set is I(f ) = {z : f n (z) as n }.

15 Complex dynamics f : C C is analytic f n is the nth iterate of f Definition The escaping set is I(f ) = {z : f n (z) as n }. Definition The Julia set J(f ) is the boundary of the escaping set.

16 Complex dynamics f : C C is analytic f n is the nth iterate of f Definition The escaping set is I(f ) = {z : f n (z) as n }. Definition The Julia set J(f ) is the boundary of the escaping set. Definition The Fatou set F(f ) = C \ J(f ).

17 Background Quadratics f c (z) = z 2 + c

18 Background Quadratics f c (z) = z 2 + c dim J(f c ) > 0

19 Background Quadratics f c (z) = z 2 + c dim J(f c ) > 0 dim J(f c ) takes every value in (0, 2]

20 Quadratic examples f (z) = z

21 Quadratic examples f (z) = z

22 Quadratic examples f (z) = z

23 Quadratic examples f (z) = z 2 2.1

24 Exponential functions f (z) = 1 4 ez

25 Exponential functions f (z) = 1 4 ez I(f ) is a Cantor bouquet of curves without some of the endpoints

26 Exponential functions f (z) = 1 4 ez I(f ) is a Cantor bouquet of curves without some of the endpoints J(f ) = I(f ) plus all endpoints

27 Exponential functions f (z) = 1 4 ez I(f ) is a Cantor bouquet of curves without some of the endpoints J(f ) = I(f ) plus all endpoints dim J(f ) = dim I(f ) = 2 (McMullen)

28 Exponential functions f (z) = 1 4 ez I(f ) is a Cantor bouquet of curves without some of the endpoints J(f ) = I(f ) plus all endpoints dim J(f ) = dim I(f ) = 2 (McMullen) curves without endpoints have dimension 1; endpoints have dimension 2 (Karpińska s paradox)

29 Possible values of dimensions Stallard,

30 Possible values of dimensions Stallard, For any d satisfying 1 < d 2, there exists f with dim J(f ) = dim I(f ) = d

31 Possible values of dimensions Stallard, For any d satisfying 1 < d 2, there exists f with dim J(f ) = dim I(f ) = d If f is in class B then dim J(f ) > 1

32 Moving up the ladder Open University Tim born, 9 months maternity leave, switched to working 0.5

33 Moving up the ladder Open University Tim born, 9 months maternity leave, switched to working won LMS Whitehead Prize

34 Moving up the ladder Open University Tim born, 9 months maternity leave, switched to working won LMS Whitehead Prize Abigail born, 9 months maternity leave

35 Moving up the ladder Open University Tim born, 9 months maternity leave, switched to working won LMS Whitehead Prize Abigail born, 9 months maternity leave Promoted to Senior Lecturer Began to collaborate with Phil Rippon on the structure of the escaping set

36 Moving up the ladder Open University Tim born, 9 months maternity leave, switched to working won LMS Whitehead Prize Abigail born, 9 months maternity leave Promoted to Senior Lecturer Began to collaborate with Phil Rippon on the structure of the escaping set Increased to 0.6 working, started going to conferences again

37 Moving up the ladder Open University Tim born, 9 months maternity leave, switched to working won LMS Whitehead Prize Abigail born, 9 months maternity leave Promoted to Senior Lecturer Began to collaborate with Phil Rippon on the structure of the escaping set Increased to 0.6 working, started going to conferences again Increased to 0.7 working Increased to 0.8 working

38 Moving up the ladder Open University Tim born, 9 months maternity leave, switched to working won LMS Whitehead Prize Abigail born, 9 months maternity leave Promoted to Senior Lecturer Began to collaborate with Phil Rippon on the structure of the escaping set Increased to 0.6 working, started going to conferences again Increased to 0.7 working Increased to 0.8 working Promoted to Professor

39 Moving up the ladder Open University Tim born, 9 months maternity leave, switched to working won LMS Whitehead Prize Abigail born, 9 months maternity leave Promoted to Senior Lecturer Began to collaborate with Phil Rippon on the structure of the escaping set Increased to 0.6 working, started going to conferences again Increased to 0.7 working Increased to 0.8 working Promoted to Professor Increased to full-time working

40 The structure of the escaping set Eremenko s conjectures Eremenko conjectured (1989) that, if f is transcendental entire, then 1. All components of I(f ) are unbounded 2. I(f ) consists of curves to.

41 The structure of the escaping set Eremenko s conjectures Eremenko conjectured (1989) that, if f is transcendental entire, then 1. All components of I(f ) are unbounded 2. I(f ) consists of curves to. Theorem (Rottenfusser, Rückert, Rempe and Schleicher) I(f ) consists of curves to if f is a finite composition of functions of finite order in class B.

42 The structure of the escaping set Eremenko s conjectures Eremenko conjectured (1989) that, if f is transcendental entire, then 1. All components of I(f ) are unbounded 2. I(f ) consists of curves to. Theorem (Rottenfusser, Rückert, Rempe and Schleicher) I(f ) consists of curves to if f is a finite composition of functions of finite order in class B. RRRS shows that there is a function in class B for which all path-connected components of I(f ) are points; all components of I(f ) are unbounded.

43 The fast escaping set Let r > 0 and M(r) = max z =r f (z).

44 The fast escaping set Let r > 0 and M(r) = max z =r f (z). Definition (Bergweiler and Hinkkanen, 1999) The fast escaping set is A(f ) = L N f L (A R (f )) where: A R (f ) = {z C : f n (z) M n (R) n N}, where R > 0 is such that M n (R) as n.

45 The fast escaping set Let r > 0 and M(r) = max z =r f (z). Definition (Bergweiler and Hinkkanen, 1999) The fast escaping set is A(f ) = L N f L (A R (f )) where: A R (f ) = {z C : f n (z) M n (R) n N}, where R > 0 is such that M n (R) as n. Theorem (follows from Eremenko, 1989) J(f ) = A(f )

46 The fast escaping set Let r > 0 and M(r) = max z =r f (z). Definition (Bergweiler and Hinkkanen, 1999) The fast escaping set is A(f ) = L N f L (A R (f )) where: A R (f ) = {z C : f n (z) M n (R) n N}, where R > 0 is such that M n (R) as n. Theorem (follows from Eremenko, 1989) J(f ) = A(f ) Theorem (Rippon and Stallard, 2005) All components of A(f ) are unbounded and hence I(f ) has at least one unbounded component.

47 A new structure for the escaping set Rippon and Stallard Theorem If f has a multiply connected Fatou component

48 A new structure for the escaping set Rippon and Stallard Theorem If f has a multiply connected Fatou component then A(f ) and I(f ) are connected and are spiders webs.

49 A new structure for the escaping set Rippon and Stallard Theorem If f has a multiply connected Fatou component then A(f ) and I(f ) are connected and are spiders webs. Definition E is a spider s web if E is connected; there is a sequence of bounded simply connected domains G n with G n E, G n+1 G n, G n = C. n N

50 Picture of a spider s web f (z) = 0.5(cos z 1/4 + cosh z 1/4 )

51 Properties of spiders webs Rippon and Stallard Theorem If A R (f ) is a spider s web then I(f ) is a spider s web and so is connected;

52 Properties of spiders webs Rippon and Stallard Theorem If A R (f ) is a spider s web then I(f ) is a spider s web and so is connected; each point in A(f ) is the limit of a sequence of points, each in a distinct component of A(f ) c ;

53 Properties of spiders webs Rippon and Stallard Theorem If A R (f ) is a spider s web then I(f ) is a spider s web and so is connected; each point in A(f ) is the limit of a sequence of points, each in a distinct component of A(f ) c ; there are no unbounded Fatou components.

54 Surprising link with Baker s conjecture Baker s conjecture If f has order less than 1/2 then f has no unbounded Fatou components.

55 Surprising link with Baker s conjecture Baker s conjecture If f has order less than 1/2 then f has no unbounded Fatou components.

56 Surprising link with Baker s conjecture Baker s conjecture If f has order less than 1/2 then f has no unbounded Fatou components. Partial results were proved by using lemmas that imply that A R (f ) is a spider s web (and hence no unbounded Fatou components and I(f ) is connected).

57 Surprising link with Baker s conjecture Baker s conjecture If f has order less than 1/2 then f has no unbounded Fatou components. Partial results were proved by using lemmas that imply that A R (f ) is a spider s web (and hence no unbounded Fatou components and I(f ) is connected). Funded by EPSRC (0.5FT, , ) to investigate the link between Baker s conjecture and Eremenko s conjecture.