Analytic Continuation of Analytic (Fractal) Functions

Transcription

1 Analytic Continuation of Analytic (Fractal) Functions Michael F. Barnsley Andrew Vince (UFL) Australian National University 10 December 2012

2 Analytic continuations of fractals generalises analytic continuation of analytic functions Given a (piece of) an analytic fractal function can we continue it?

3 Analytic continuations of fractals generalises analytic continuation of analytic functions Given a (piece of) an analytic fractal function can we continue it? Analytic continuation is a cornerstone notion in mathematics and applications

4 Analytic iterated function systems (IFS) W = {X; w 1, w 2,.., w N } where w n : X X real analytic X =complete subset of R 2

5 Analytic iterated function systems (IFS) W = {X; w 1, w 2,.., w N } where w n : X X real analytic X =complete subset of R 2 W : 2 X 2 X defined by W(S) = w i (S);

6 Analytic iterated function systems (IFS) W = {X; w 1, w 2,.., w N } where w n : X X real analytic X =complete subset of R 2 W : 2 X 2 X defined by W(S) = w i (S); an attractor is a compact set A K s.t. lim k W k (S) = A for all bounded nonempty S X

7 analytic fractal functions W = {X R 2 ; w 1, w 2,.., w N } where w n : X X analytic f : [0, 1] R is an analytic fractal function if

8 analytic fractal functions W = {X R 2 ; w 1, w 2,.., w N } where w n : X X analytic f : [0, 1] R is an analytic fractal function if G(f ) is the attractor of an analytic IFS

9 analytic fractal functions W = {X R 2 ; w 1, w 2,.., w N } where w n : X X analytic f : [0, 1] R is an analytic fractal function if G(f ) is the attractor of an analytic IFS examples include analytic functions, arcs of wavelets, fractal interpolation functions, Wieirstrass nowhere differentiable functions, and functions defined by Julia sets.

10 Analytic fractal functions

11 Analytic fractal functions

12 All real analytic functions are analytic fractal functions Theorem 1: If f : [0, 1] R is real analytic on a neighborhood of [0, 1] then there is an analytic IFS whose attractor is G(f ).

13 All real analytic functions are analytic fractal functions Theorem 1: If f : [0, 1] R is real analytic on a neighborhood of [0, 1] then there is an analytic IFS whose attractor is G(f ). PROOF: If f is strictly increasing with f (x) not varying too much, IFS is {X; w 1 = ( x 2, f (f 1 (y) 2 )), w 2 = (x/2 + 1/2, f ( f 1 (y) + 1 ))}. 2

14 All real analytic functions are analytic fractal functions Theorem 1: If f : [0, 1] R is real analytic on a neighborhood of [0, 1] then there is an analytic IFS whose attractor is G(f ). PROOF: If f is strictly increasing with f (x) not varying too much, IFS is {X; w 1 = ( x 2, f (f 1 (y) 2 )), w 2 = (x/2 + 1/2, f ( f 1 (y) + 1 ))}. 2 Otherwise, first conjugate by T(x, y) = (x, y + Cx) with C sufficiently large.

15 Examples: QUADRATIC, EXPONENTIAL The graph of f : [0, 1] R, f (x) = x 2 is the attractor of the analytic IFS {R 2 ; w 1 = (x/2, y/4), w 2 = ((x + 1)/2, (2x + y + 1)/4)}

16 Examples: QUADRATIC, EXPONENTIAL The graph of f : [0, 1] R, f (x) = x 2 is the attractor of the analytic IFS {R 2 ; w 1 = (x/2, y/4), w 2 = ((x + 1)/2, (2x + y + 1)/4)} The graph of f : [1, 2] R, f (x) = e x is the attractor of the analytic IFS {X; w 1 = (x/2 + 1/2, ey), w 2 (x, y) = (x/2 + 1, e y)}

17 fractal continuation=fractal "blow-up" Let A be attractor of W = {X; w 1, w 2 }

18 fractal continuation=fractal "blow-up" Let A be attractor of W = {X; w 1, w 2 } Let θ = θ 1 θ 2... {1, 2}

19 fractal continuation=fractal "blow-up" Let A be attractor of W = {X; w 1, w 2 } Let θ = θ 1 θ 2... {1, 2} B(θ 1 θ 2...θ k ) = w 1 θ 1 w 1 θ 2... w 1 (A) θ k

20 fractal continuation=fractal "blow-up" Let A be attractor of W = {X; w 1, w 2 } Let θ = θ 1 θ 2... {1, 2} B(θ 1 θ 2...θ k ) = w 1 θ 1 w 1 θ 2... w 1 θ k (A) A = B( ) B(θ 1 ) B(θ 1 θ 2 )... B(θ 1 θ 2...θ k )...

21 fractal continuation=fractal "blow-up" Let A be attractor of W = {X; w 1, w 2 } Let θ = θ 1 θ 2... {1, 2} B(θ 1 θ 2...θ k ) = w 1 θ 1 w 1 θ 2... w 1 θ k (A) A = B( ) B(θ 1 ) B(θ 1 θ 2 )... B(θ 1 θ 2...θ k )... B(θ) := k N B(θ 1 θ 2...θ k ) is a continuation of A.

22 Continuation of real analytic attractors Let A be attractor of an analytic IFS W = {X; w 1, w 2 }.

23 Continuation of real analytic attractors Let A be attractor of an analytic IFS W = {X; w 1, w 2 }. Theorem 2: If A is an arc of the graph G(f ) of a real analytic function f : D R R, then B(θ) is the graph of an analytic continuation of f.

24 Example (i) PARABOLA

25 Example (i) PARABOLA The graph of f : [0, 1] R, f (x) = x 2 is the attractor of the (analytic) IFS W ={R 2 ; w 1 = (x/2, y/4), w 2 = ((x + 1)/2, (2x + y + 1)/4)}

26 Example (i) PARABOLA The graph of f : [0, 1] R, f (x) = x 2 is the attractor of the (analytic) IFS W ={R 2 ; w 1 = (x/2, y/4), w 2 = ((x + 1)/2, (2x + y + 1)/4)} For almost all θ I, B(θ) = {(x, x 2 ) : x R}

27 Examples: real analytic attractors Example (ii) EXPONENTIAL

28 Examples: real analytic attractors Example (ii) EXPONENTIAL The graph of f : [1, 2] R, f (x) = e x is the attractor of the IFS W ={R 2 ; w 1 = (x/2 + 1/2, ey), w 2 (x, y) = (x/2 + 1, e 2 y)}

29 Examples: real analytic attractors Example (ii) EXPONENTIAL The graph of f : [1, 2] R, f (x) = e x is the attractor of the IFS W ={R 2 ; w 1 = (x/2 + 1/2, ey), w 2 (x, y) = (x/2 + 1, e 2 y)} For almost all θ I, B(θ) = {(x, e x ) : x R}

30 Description of FIFs x 0 = 0 < x 1 < x 2 = 1, (x i, y i ) R 2 : i = 0, 1, 2

31 Description of FIFs x 0 = 0 < x 1 < x 2 = 1, (x i, y i ) R 2 : i = 0, 1, 2 w i : X X

32 Description of FIFs x 0 = 0 < x 1 < x 2 = 1, (x i, y i ) R 2 : i = 0, 1, 2 w i : X X w i (x, y) = (a i x + h i, F i (x, y)) with w i (x 0, y 0 ) = (x i 1, y i 1 ), w i (x N, y N ) = (x i, y i )

33 Description of FIFs x 0 = 0 < x 1 < x 2 = 1, (x i, y i ) R 2 : i = 0, 1, 2 w i : X X w i (x, y) = (a i x + h i, F i (x, y)) with w i (x 0, y 0 ) = (x i 1, y i 1 ), w i (x N, y N ) = (x i, y i ) F i (x, y)/ y < 1

34 Description of FIFs x 0 = 0 < x 1 < x 2 = 1, (x i, y i ) R 2 : i = 0, 1, 2 w i : X X w i (x, y) = (a i x + h i, F i (x, y)) with w i (x 0, y 0 ) = (x i 1, y i 1 ), w i (x N, y N ) = (x i, y i ) F i (x, y)/ y < 1 Theorem 0: The IFS W = {X; w 1, w 2 } has a unique attractor which is the graph of a function f : [0, 1] R.

35 Description of FIFs x 0 = 0 < x 1 < x 2 = 1, (x i, y i ) R 2 : i = 0, 1, 2 w i : X X w i (x, y) = (a i x + h i, F i (x, y)) with w i (x 0, y 0 ) = (x i 1, y i 1 ), w i (x N, y N ) = (x i, y i ) F i (x, y)/ y < 1 Theorem 0: The IFS W = {X; w 1, w 2 } has a unique attractor which is the graph of a function f : [0, 1] R. PROOF: Simple generalisation of theorem in Massopust book.

36 Description of FIFs x 0 = 0 < x 1 < x 2 = 1, (x i, y i ) R 2 : i = 0, 1, 2 w i : X X w i (x, y) = (a i x + h i, F i (x, y)) with w i (x 0, y 0 ) = (x i 1, y i 1 ), w i (x N, y N ) = (x i, y i ) F i (x, y)/ y < 1 Theorem 0: The IFS W = {X; w 1, w 2 } has a unique attractor which is the graph of a function f : [0, 1] R. PROOF: Simple generalisation of theorem in Massopust book. If the w i s are real analytic then f is called an analytic FIF.

37 Analytic FIF may be rough, or rectifiable, or smooth, or analytic

38 Continuation of real analytic attractors Theorem 2: If A is an arc of the graph of an analytic FIF f : D R R, then B(θ) is the graph of f θ : I θ R, which coincides with f on D. Moreover, dim H (G(f )) = dim H ( B(θ 1 θ 2...θ k ) for all k.

39 Example of continuation of FIFs A= attractor of {R 2 ; w 1 = (x/2,.5x + py), w 2 = (x/2 + 1,.5x + py + 1)}

40 Example of continuation of FIFs A= attractor of {R 2 ; w 1 = (x/2,.5x + py), w 2 = (x/2 + 1,.5x + py + 1)}...is the graph of a FIF: (Minkowski) dimension D(d), cts. dep. on p ( 1, 1)

41 Example of continuation of FIFs A= attractor of {R 2 ; w 1 = (x/2,.5x + py), w 2 = (x/2 + 1,.5x + py + 1)}...is the graph of a FIF: (Minkowski) dimension D(d), cts. dep. on p ( 1, 1) here p = 0.3 and the graph is of finite length

42 Example of continuation of FIFs same as the previous example, but now p = 0.5

43 Example of continuation of FIFs same as the previous example, but now p = 0.5 once differentiable except at countably many points

44 Example: fractal continuations Fractal continuations of a high Minkowski dimension analytic fractal function

45 Example: fractal continuations Fractal continuations of a high Minkowski dimension analytic fractal function nowhere differentiable and dim( B(θ)) = dim(a) > 1

46 Fractal continuations Fractal continuations of an analytic fractal function

47 Examples: differentiable functions EXAMPLE : continuation of the attractor of two affine maps: possesses continuous first derivative.

48 Examples: differentiable functions EXAMPLE : continuation of the attractor of two affine maps: possesses continuous first derivative. Related to the Kigami triangle.

49 Examples: julia sets EXAMPLE (vi): Part of the Julia set J for z Because of symmetries, also the attractor of {D C; z + 0.3, z 0.3}.

50 Examples: julia sets continued Continuation of EXAMPLE: in this example B is a quasi-circle. Continuation is unique, equals the full Julia set.

51 Examples: julia sets continued

52 Examples: Daubechies wavelets These wavelets are piecewise fractal interpolation functions, attractors of analytic IFSs. Pieces have unique analytic continuations B(θ)

53 Uniqueness of Fractal Continuations Theorem 4 If the graph G(f ) of f : [0, 1] R is the (FIF) attractor of two different analytic IFSs, each of the form {R 2 ; (a n x + h n, F n (x, y)) n = 1, 2,..., N}, then the continuation associated with is same for each.

54 Uniqueness of Fractal Continuations Theorem 4 If the graph G(f ) of f : [0, 1] R is the (FIF) attractor of two different analytic IFSs, each of the form {R 2 ; (a n x + h n, F n (x, y)) n = 1, 2,..., N}, then the continuation associated with is same for each. PROOF: Commutation and Wieirstass preparation theorem.

55 Uniqueness of Fractal Continuations Theorem 5 If the graph G(f ) of f : [0, 1] R is a (FIF) attractor of an analytic IFS of the form {R 2, (l n (x), F n (x, y), N = 2}, and f (x) is infinitely differentiable, or G(f ) has a tangent at all except countably many points, then the set of continuations is unique.

56 Uniqueness of Fractal Continuations Theorem 5 If the graph G(f ) of f : [0, 1] R is a (FIF) attractor of an analytic IFS of the form {R 2, (l n (x), F n (x, y), N = 2}, and f (x) is infinitely differentiable, or G(f ) has a tangent at all except countably many points, then the set of continuations is unique. PROOF:The main steps:

57 Uniqueness of Fractal Continuations Theorem 5 If the graph G(f ) of f : [0, 1] R is a (FIF) attractor of an analytic IFS of the form {R 2, (l n (x), F n (x, y), N = 2}, and f (x) is infinitely differentiable, or G(f ) has a tangent at all except countably many points, then the set of continuations is unique. PROOF:The main steps: (i) Theorem A: f is Lipshitz (bootstrap argument).

58 Uniqueness of Fractal Continuations Theorem 5 If the graph G(f ) of f : [0, 1] R is a (FIF) attractor of an analytic IFS of the form {R 2, (l n (x), F n (x, y), N = 2}, and f (x) is infinitely differentiable, or G(f ) has a tangent at all except countably many points, then the set of continuations is unique. PROOF:The main steps: (i) Theorem A: f is Lipshitz (bootstrap argument). (ii) Theorem B: if G(f ) is rectifiable then f (x) is discontinuous only at points with two addresses.

59 Uniqueness of Fractal Continuations Theorem 5 If the graph G(f ) of f : [0, 1] R is a (FIF) attractor of an analytic IFS of the form {R 2, (l n (x), F n (x, y), N = 2}, and f (x) is infinitely differentiable, or G(f ) has a tangent at all except countably many points, then the set of continuations is unique. PROOF:The main steps: (i) Theorem A: f is Lipshitz (bootstrap argument). (ii) Theorem B: if G(f ) is rectifiable then f (x) is discontinuous only at points with two addresses. (iii) If two IFSs, Theorem B implies set of points with double addresses is same.

60 Uniqueness of Fractal Continuations Theorem 5 If the graph G(f ) of f : [0, 1] R is a (FIF) attractor of an analytic IFS of the form {R 2, (l n (x), F n (x, y), N = 2}, and f (x) is infinitely differentiable, or G(f ) has a tangent at all except countably many points, then the set of continuations is unique. PROOF:The main steps: (i) Theorem A: f is Lipshitz (bootstrap argument). (ii) Theorem B: if G(f ) is rectifiable then f (x) is discontinuous only at points with two addresses. (iii) If two IFSs, Theorem B implies set of points with double addresses is same. (iv) Theorem A implies interpolation nodes are the same. Then Wieirstrass.

61 Uniqueness of Fractal Continuations Theorem 5 If the graph G(f ) of f : [0, 1] R is a (FIF) attractor of an analytic IFS of the form {R 2, (l n (x), F n (x, y), N = 2}, and f (x) is infinitely differentiable, or G(f ) has a tangent at all except countably many points, then the set of continuations is unique. PROOF:The main steps: (i) Theorem A: f is Lipshitz (bootstrap argument). (ii) Theorem B: if G(f ) is rectifiable then f (x) is discontinuous only at points with two addresses. (iii) If two IFSs, Theorem B implies set of points with double addresses is same. (iv) Theorem A implies interpolation nodes are the same. Then Wieirstrass. arxiv: v2 (Nov 2012)

62 Uniqueness of Fractal Continuations Theorem 5 If the graph G(f ) of f : [0, 1] R is a (FIF) attractor of an analytic IFS of the form {R 2, (l n (x), F n (x, y), N = 2}, and f (x) is infinitely differentiable, or G(f ) has a tangent at all except countably many points, then the set of continuations is unique. PROOF:The main steps: (i) Theorem A: f is Lipshitz (bootstrap argument). (ii) Theorem B: if G(f ) is rectifiable then f (x) is discontinuous only at points with two addresses. (iii) If two IFSs, Theorem B implies set of points with double addresses is same. (iv) Theorem A implies interpolation nodes are the same. Then Wieirstrass. arxiv: v2 (Nov 2012) Conjecture: continuations of analytic FIF is unique.

63 Fast basins the fast-basin is B := {x X : k N s.t.w k ({x}) A = } = θ {1,2} B(θ)

64 Fast basins the fast-basin is B := {x X : k N s.t.w k ({x}) A = } = θ {1,2} B(θ) Reverse invariance: W 1 ( B) = A W 1 ( B)

65 Picture shows fast basin for Sierpinski

66 Inherited properties Theorem 6 (with K. Lesniak) The following properties are inherited from A to B and B(θ): (i) (arc-) connected; (ii) empty interior; (iii) σ-porous (*); (iv) topological covering dimension; (v) Hausdorff dimension (*).

67 Inherited properties Theorem 6 (with K. Lesniak) The following properties are inherited from A to B and B(θ): (i) (arc-) connected; (ii) empty interior; (iii) σ-porous (*); (iv) topological covering dimension; (v) Hausdorff dimension (*). (*) means : assumes W is bi-lipshitz.

68 Branched) fracta(l-mani)folds (

69 Fractal manifolds fractal manifold using fern

70 Fast basins and fractal manifolds

71 Kigami fast-basin

72 Kigami fast-basin

73 Kigami fast-basin

74 Kigami fast-basin

75 Kigami fractal manifold