slits, spirals and loewner hulls =4 t 0 - mathematicspietro/seattle19-slides/joan-lind.pdf · joan...
TRANSCRIPT
![Page 1: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/1.jpg)
Slits, spirals and Loewner hulls
1 Introduction and Results
Let λ(t) be continuous and real valued and let gt : H \ Kt → H be the solution to the Loewnerequation
d
dtgt(z) =
2
gt(z) − λ(t), g0(z) = z ∈ H, (1.1)
where H is the upper half-plane. It was shown in [MR1] and [Li] that if λ is Holder continuouswith exponent 1/2 and if ||λ||1/2 < 4, then there is a simple curve γ with γ[0, t] = Kt and
γ \ γ(0) ⊂ H. The norm 4 is sharp as the examples λ(t) = κ√
1 − t show: Indeed, by [KNK],γ touches back on the real line if κ ≥ 4 (hence the driving term λ(t) = κ for 0 ≤ t ≤ t0 andλ(t) = κ
√t0 + 1 − t for t0 ≤ t ≤ t0 + 1 has a self-intersection in H for t0 sufficiently large).
It was also shown in [MR1] that there is a λ with ||λ||1/2 < ∞ such that K1 spirals infinitelyoften around some disc, and hence is not locally connected. The starting point of this paperis the observation that from the conformal mapping point of view, the zero angle cusp at thetangential self-intersection for λ(t) = 4
√1 − t is very similar to the infinitely spiraling prime
end, and that this is reflected in the driving terms:
Theorem 1.1. If γ is a sufficiently smooth infinite spiral of half-plane capacity T , or if γ hasa tangential self-intersection, then its driving term λ satisfies
limt→T
|λ(T ) − λ(t)|√T − t
= 4.
Figure 1: An infinite spiral converging towards a star.
See Sections 2.1 and 5 for the definitions and precise statements. In Section 5 we show that forevery compact connected set A ⊂ H with connected complement, there is a sufficiently smoothinfinite spiral winding infinitely often around A with limit set ∂A; see Figure 1. The followingnatural question has been asked by Omer Angel: If the hull of λ is generated by a continuouscurve γ and if r < 1, is it true that the hull of rλ is generated by a continuous curve, too? Inother words, is the space of driving terms of continuous curves starlike? We answer this questionin the negative by proving
Theorem 1.2. If γ is a sufficiently smooth infinite spiral of half-plane capacity T , and if λ isits driving term, then the trace of rλ is continuous on the closed interval [0, T ] for all r = ±1.
The main work is in proving a form of stability of the (nontangential) self-intersection of λ(t) =κ√
1 − t for κ > 4 :
2
Joan Lind
University of Tennessee
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The story begins
Bieberbach conjecture (1916):For f (z) = z + a2z
2 + a3z3 + · · · conformal on D, then |an| ≤ n.
Charles Loewner introduced the Loewner equation in 1923 to provethe n = 3 case.
Louis des Branges again used the Loewner equation when heproved the conjecture in 1985.
![Page 3: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/3.jpg)
The story begins
Bieberbach conjecture (1916):For f (z) = z + a2z
2 + a3z3 + · · · conformal on D, then |an| ≤ n.
Charles Loewner introduced the Loewner equation in 1923 to provethe n = 3 case.
Louis des Branges again used the Loewner equation when heproved the conjecture in 1985.
![Page 4: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/4.jpg)
The story begins
Bieberbach conjecture (1916):For f (z) = z + a2z
2 + a3z3 + · · · conformal on D, then |an| ≤ n.
Charles Loewner introduced the Loewner equation in 1923 to provethe n = 3 case.
Louis des Branges again used the Loewner equation when heproved the conjecture in 1985.
![Page 5: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/5.jpg)
The story continues
0
Simple Random Walk (SRW) in 2 dimensions– has a conformally invariant scaling limit: 2-d Brownian motion
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The story continues
Schramm’s question: Do other 2-dimensional random walks (suchas SAW, LERW, etc.) have conformally invariant scaling limits?
0
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The story continues
In 2000, Oded Schramm introduced SLEκ, a family of randomprocesses that contain the possible conformally invariant scalinglimits.
Through the Loewner equation, SLEκ correspond to√κBt .
![Page 8: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/8.jpg)
The story continues
In 2000, Oded Schramm introduced SLEκ, a family of randomprocesses that contain the possible conformally invariant scalinglimits.
Through the Loewner equation, SLEκ correspond to√κBt .
![Page 9: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/9.jpg)
The Loewner equation
L Equ
!
1
growing families of 2-d sets real-valued functions
![Page 10: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/10.jpg)
From curves to functions
�(t)
�(t)
�(t)
gt
0 �(t)
1
![Page 11: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/11.jpg)
From curves to functions
�(t)
�(t)
�(t)
gt
0 �(t)
1
![Page 12: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/12.jpg)
From curves to functions
�(t)
�(t)
�(t)
gt
�(t)
1
![Page 13: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/13.jpg)
From curves to functions
�(t)
�(t)
�(t)
gt
�(t)
1
gt(z) = z +ctz
+ O
(1
z2
)for z near infinity
![Page 14: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/14.jpg)
From curves to functions
�(t)
�(t)
�(t)
gt
�(t)
1
gt(z) = z +2t
z+ O
(1
z2
)for z near infinity
![Page 15: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/15.jpg)
From curves to functions
�(t)
�(t)
�(t)
gt
�(t)
1
∂
∂tgt(z) =
2
gt(z)− λ(t)
![Page 16: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/16.jpg)
From functions to growing families of sets
For a continuous, real-valued function λ(t) and z ∈ H, consider
∂
∂tgt(z) =
2
gt(z)− λ(t), g0(z) = z
Loewner hulls: Kt = {z ∈ H : gs(z) = λ(s) for some s ≤ t}.
Theorem: gt is a conformal map from H \ Kt onto H.
![Page 17: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/17.jpg)
From functions to growing families of sets
For a continuous, real-valued function λ(t) and z ∈ H, consider
∂
∂tgt(z) =
2
gt(z)− λ(t), g0(z) = z
Loewner hulls: Kt = {z ∈ H : gs(z) = λ(s) for some s ≤ t}.
Theorem: gt is a conformal map from H \ Kt onto H.
![Page 18: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/18.jpg)
From functions to growing families of sets
For a continuous, real-valued function λ(t) and z ∈ H, consider
∂
∂tgt(z) =
2
gt(z)− λ(t), g0(z) = z
Loewner hulls: Kt = {z ∈ H : gs(z) = λ(s) for some s ≤ t}.
Theorem: gt is a conformal map from H \ Kt onto H.
![Page 19: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/19.jpg)
Loewner flow
∂
∂tgt(z) = 2
Re gt(z)− λ(t)
|gt(z)− λ(t)|2 − 2iIm gt(z)
|gt(z)− λ(t)|2
0.2
0.4
0.6
0.8
1
–1 –0.5 0.5 1
![Page 20: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/20.jpg)
Example
Loewner hull generated by λ(t) ≡ 0.
![Page 21: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/21.jpg)
Loewner equation
L Equ
!
1
growing families of 2-d sets real-valued functions
![Page 22: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/22.jpg)
The geometry of SLE curves
L Equ
SLEκ ←→ √κBt
0 ≤ κ ≤ 4 4 < κ < 8 8 ≤ κ
![Page 23: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/23.jpg)
The geometry of SLE curves
L Equ
SLEκ ←→ √κBt
0 ≤ κ ≤ 4 4 < κ < 8 8 ≤ κ
![Page 24: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/24.jpg)
The geometry of SLE curves
SLE2 SLE6
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Simple curve Loewner hulls
Question: When does the deterministic Loewner equation generatea simple curve?
Lip(1/2) functions:
|λ(t)− λ(s)| ≤ M |t − s|1/2
for all t, s is the domain of λ. The smallest such M is ||λ||1/2.
Answer: (Marshall, Rohde) There exists C0 > 0 so that forλ ∈ Lip(1/2) with ||λ||1/2 < C0, then the Loewner hull is aquasislit γ.
![Page 26: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/26.jpg)
Simple curve Loewner hulls
Question: When does the deterministic Loewner equation generatea simple curve?
Lip(1/2) functions:
|λ(t)− λ(s)| ≤ M |t − s|1/2
for all t, s is the domain of λ. The smallest such M is ||λ||1/2.
Answer: (Marshall, Rohde) There exists C0 > 0 so that forλ ∈ Lip(1/2) with ||λ||1/2 < C0, then the Loewner hull is aquasislit γ.
![Page 27: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/27.jpg)
Simple curve Loewner hulls
Question: When does the deterministic Loewner equation generatea simple curve?
Lip(1/2) functions:
|λ(t)− λ(s)| ≤ M |t − s|1/2
for all t, s is the domain of λ. The smallest such M is ||λ||1/2.
Answer: (Marshall, Rohde) There exists C0 > 0 so that forλ ∈ Lip(1/2) with ||λ||1/2 < C0, then the Loewner hull is aquasislit γ.
![Page 28: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/28.jpg)
Simple curve Loewner hulls
Question: Do all Lip(1/2) driving functions generate simple curves?
Answer: (Marshall, Rohde) No. There is a non-simple example (acurve that spirals around a disc) that is generated by a Lip(1/2)driving function.
![Page 29: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/29.jpg)
Simple curve Loewner hulls
Question: Do all Lip(1/2) driving functions generate simple curves?
Answer: (Marshall, Rohde) No. There is a non-simple example (acurve that spirals around a disc) that is generated by a Lip(1/2)driving function.
![Page 30: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/30.jpg)
Simple curve Loewner hulls
Question: What is the optimal value of C0 for the Marshall-Rohdetheorem?
Answer: (L) C0 = 4.
Key examples: λ(t) = −c√
1− t
c = 3 c = 5
![Page 31: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/31.jpg)
Simple curve Loewner hulls
Question: What is the optimal value of C0 for the Marshall-Rohdetheorem?
Answer: (L) C0 = 4.
Key examples: λ(t) = −c√
1− t
c = 3 c = 5
![Page 32: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/32.jpg)
Simple curve Loewner hulls
Question: What is the optimal value of C0 for the Marshall-Rohdetheorem?
Answer: (L) C0 = 4.
Key examples: λ(t) = −c√
1− t
c = 3 c = 5
![Page 33: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/33.jpg)
Loewner deformations
Omer Angel’s question:For r < 1, what can you say about Loewner hull driven by rλ?
Is it at least as nice as the hull driven by λ?
This is true for SLE.
0 ≤ κ ≤ 4 4 < κ < 8 8 ≤ κ
![Page 34: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/34.jpg)
Loewner deformations
Omer Angel’s question:For r < 1, what can you say about Loewner hull driven by rλ?Is it at least as nice as the hull driven by λ?
This is true for SLE.
0 ≤ κ ≤ 4 4 < κ < 8 8 ≤ κ
![Page 35: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/35.jpg)
Loewner deformations
Omer Angel’s question:For r < 1, what can you say about Loewner hull driven by rλ?Is it at least as nice as the hull driven by λ?
This is true for SLE.
0 ≤ κ ≤ 4 4 < κ < 8 8 ≤ κ
![Page 36: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/36.jpg)
Loewner deformations
Omer Angel’s question:For r < 1, what can you say about Loewner hull driven by rλ?Is it at least as nice as the hull driven by λ?
This is true for λ(t) = −c√
1− t.
c = 3 c = 5
![Page 37: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/37.jpg)
Loewner deformations
Omer Angel’s question:For r < 1, what can you say about Loewner hull driven by rλ?Is it at least as nice as the hull driven by λ?
Is it always true?
Answer: N0.
![Page 38: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/38.jpg)
Loewner deformations
Omer Angel’s question:For r < 1, what can you say about Loewner hull driven by rλ?Is it at least as nice as the hull driven by λ?
Is it always true?
Answer: N0.
![Page 39: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/39.jpg)
Loewner deformations
1 Introduction and Results
Let λ(t) be continuous and real valued and let gt : H \ Kt → H be the solution to the Loewnerequation
d
dtgt(z) =
2
gt(z) − λ(t), g0(z) = z ∈ H, (1.1)
where H is the upper half-plane. It was shown in [MR1] and [Li] that if λ is Holder continuouswith exponent 1/2 and if ||λ||1/2 < 4, then there is a simple curve γ with γ[0, t] = Kt and
γ \ γ(0) ⊂ H. The norm 4 is sharp as the examples λ(t) = κ√
1 − t show: Indeed, by [KNK],γ touches back on the real line if κ ≥ 4 (hence the driving term λ(t) = κ for 0 ≤ t ≤ t0 andλ(t) = κ
√t0 + 1 − t for t0 ≤ t ≤ t0 + 1 has a self-intersection in H for t0 sufficiently large).
It was also shown in [MR1] that there is a λ with ||λ||1/2 < ∞ such that K1 spirals infinitelyoften around some disc, and hence is not locally connected. The starting point of this paperis the observation that from the conformal mapping point of view, the zero angle cusp at thetangential self-intersection for λ(t) = 4
√1 − t is very similar to the infinitely spiraling prime
end, and that this is reflected in the driving terms:
Theorem 1.1. If γ is a sufficiently smooth infinite spiral of half-plane capacity T , or if γ hasa tangential self-intersection, then its driving term λ satisfies
limt→T
|λ(T ) − λ(t)|√T − t
= 4.
Figure 1: An infinite spiral converging towards a star.
See Sections 2.1 and 5 for the definitions and precise statements. In Section 5 we show that forevery compact connected set A ⊂ H with connected complement, there is a sufficiently smoothinfinite spiral winding infinitely often around A with limit set ∂A; see Figure 1. The followingnatural question has been asked by Omer Angel: If the hull of λ is generated by a continuouscurve γ and if r < 1, is it true that the hull of rλ is generated by a continuous curve, too? Inother words, is the space of driving terms of continuous curves starlike? We answer this questionin the negative by proving
Theorem 1.2. If γ is a sufficiently smooth infinite spiral of half-plane capacity T , and if λ isits driving term, then the trace of rλ is continuous on the closed interval [0, T ] for all r = ±1.
The main work is in proving a form of stability of the (nontangential) self-intersection of λ(t) =κ√
1 − t for κ > 4 :
2
Theorem (L, Marshall, Rohde)
Let λ be the driving function for the star spiral. When 0 < r < 1,then rλ generates a simple curve, and when r > 1, then rλgenerates a “bubble” curve.
![Page 40: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/40.jpg)
Spirals and collisions
Theorem (L, Marshall, Rohde)
If γ is a suffiently nice infinite spiral of half-plane capacity T , or ifγ has a tangential self-intersection, then its driving term λ satisfies
limt→T
|λ(T )− λ(t)|√T − t
= 4.
Theorem (L, Marshall, Rohde)
If λ : [0,T ]→ R is sufficiently regular on [0,T ) and if
limt→T
|λ(T )− λ(t)|√T − t
= κ > 4,
then γ(T ) = limt→T
γ(t) exists, is real, and γ intersects R in the
same angle as the trace for κ√
1− t.
![Page 41: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/41.jpg)
Spirals and collisions
Theorem (L, Marshall, Rohde)
If γ is a suffiently nice infinite spiral of half-plane capacity T , or ifγ has a tangential self-intersection, then its driving term λ satisfies
limt→T
|λ(T )− λ(t)|√T − t
= 4.
Theorem (L, Marshall, Rohde)
If λ : [0,T ]→ R is sufficiently regular on [0,T ) and if
limt→T
|λ(T )− λ(t)|√T − t
= κ > 4,
then γ(T ) = limt→T
γ(t) exists, is real, and γ intersects R in the
same angle as the trace for κ√
1− t.
![Page 42: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/42.jpg)
Spirals and collisions
Scaling Property:
I If λ generates hull Kt , then cλ(t/c2) generates cKt/c2 .
Example: λ(t) = c√t
![Page 43: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/43.jpg)
Spirals and collisions
Scaling Property:
I If λ generates hull Kt , then cλ(t/c2) generates cKt/c2 .
Example: λ(t) = c√t
![Page 44: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/44.jpg)
Spirals and collisions
Concatenation Property:
I If λ generates hull Kt for t ∈ [0,T ], then λ restricted to[t0,T ] generates hull gt0(KT \ Kt0).
Kt0
λ(0)
gt0
gt0(KT \ Kt0)
λ(t0)
![Page 45: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/45.jpg)
Spirals and collisions
Concatenation Property:
I If λ generates hull Kt for t ∈ [0,T ], then λ restricted to[t0,T ] generates hull gt0(KT \ Kt0).
Kt0
λ(0)
gt0
gt0(KT \ Kt0)
λ(t0)
![Page 46: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/46.jpg)
Spirals and collisions
Start with λ = −c√
1− t.
For t0 ∈ (0, 1), map down by gt0 .
Rescale by 1/√
1− t0.
What do we obtain? The same hull that we started with.
![Page 47: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/47.jpg)
Spirals and collisions
Start with λ = −c√
1− t.
For t0 ∈ (0, 1), map down by gt0 .
Rescale by 1/√
1− t0.
What do we obtain? The same hull that we started with.
![Page 48: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/48.jpg)
Spirals and collisions
Start with λ satisfying
limt→1
|λ(1)− λ(t)|√1− t
= κ ≥ 4
Shift so that λ(1) = 0.
For t0 ∈ (0, 1), map down by gt0 .
Rescale by 1/√
1− t0.
What do we obtain? A hull that is getting closer and closer to thehull generated by ±κ
√1− t.
![Page 49: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/49.jpg)
Spirals and collisions
Start with λ satisfying
limt→1
|λ(1)− λ(t)|√1− t
= κ ≥ 4
Shift so that λ(1) = 0.
For t0 ∈ (0, 1), map down by gt0 .
Rescale by 1/√
1− t0.
What do we obtain? A hull that is getting closer and closer to thehull generated by ±κ
√1− t.
![Page 50: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/50.jpg)
Spirals and collisions
What happens when κ = 4?
4√
1− t generates a hull that intersects the real line tangentially.
The spiral behavior can be viewed as tangential behavior.
1 Introduction and Results
Let λ(t) be continuous and real valued and let gt : H \ Kt → H be the solution to the Loewnerequation
d
dtgt(z) =
2
gt(z) − λ(t), g0(z) = z ∈ H, (1.1)
where H is the upper half-plane. It was shown in [MR1] and [Li] that if λ is Holder continuouswith exponent 1/2 and if ||λ||1/2 < 4, then there is a simple curve γ with γ[0, t] = Kt and
γ \ γ(0) ⊂ H. The norm 4 is sharp as the examples λ(t) = κ√
1 − t show: Indeed, by [KNK],γ touches back on the real line if κ ≥ 4 (hence the driving term λ(t) = κ for 0 ≤ t ≤ t0 andλ(t) = κ
√t0 + 1 − t for t0 ≤ t ≤ t0 + 1 has a self-intersection in H for t0 sufficiently large).
It was also shown in [MR1] that there is a λ with ||λ||1/2 < ∞ such that K1 spirals infinitelyoften around some disc, and hence is not locally connected. The starting point of this paperis the observation that from the conformal mapping point of view, the zero angle cusp at thetangential self-intersection for λ(t) = 4
√1 − t is very similar to the infinitely spiraling prime
end, and that this is reflected in the driving terms:
Theorem 1.1. If γ is a sufficiently smooth infinite spiral of half-plane capacity T , or if γ hasa tangential self-intersection, then its driving term λ satisfies
limt→T
|λ(T ) − λ(t)|√T − t
= 4.
Figure 1: An infinite spiral converging towards a star.
See Sections 2.1 and 5 for the definitions and precise statements. In Section 5 we show that forevery compact connected set A ⊂ H with connected complement, there is a sufficiently smoothinfinite spiral winding infinitely often around A with limit set ∂A; see Figure 1. The followingnatural question has been asked by Omer Angel: If the hull of λ is generated by a continuouscurve γ and if r < 1, is it true that the hull of rλ is generated by a continuous curve, too? Inother words, is the space of driving terms of continuous curves starlike? We answer this questionin the negative by proving
Theorem 1.2. If γ is a sufficiently smooth infinite spiral of half-plane capacity T , and if λ isits driving term, then the trace of rλ is continuous on the closed interval [0, T ] for all r = ±1.
The main work is in proving a form of stability of the (nontangential) self-intersection of λ(t) =κ√
1 − t for κ > 4 :
2
![Page 51: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/51.jpg)
Spirals and collisions
What happens when κ = 4?
4√
1− t generates a hull that intersects the real line tangentially.
The spiral behavior can be viewed as tangential behavior.
1 Introduction and Results
Let λ(t) be continuous and real valued and let gt : H \ Kt → H be the solution to the Loewnerequation
d
dtgt(z) =
2
gt(z) − λ(t), g0(z) = z ∈ H, (1.1)
where H is the upper half-plane. It was shown in [MR1] and [Li] that if λ is Holder continuouswith exponent 1/2 and if ||λ||1/2 < 4, then there is a simple curve γ with γ[0, t] = Kt and
γ \ γ(0) ⊂ H. The norm 4 is sharp as the examples λ(t) = κ√
1 − t show: Indeed, by [KNK],γ touches back on the real line if κ ≥ 4 (hence the driving term λ(t) = κ for 0 ≤ t ≤ t0 andλ(t) = κ
√t0 + 1 − t for t0 ≤ t ≤ t0 + 1 has a self-intersection in H for t0 sufficiently large).
It was also shown in [MR1] that there is a λ with ||λ||1/2 < ∞ such that K1 spirals infinitelyoften around some disc, and hence is not locally connected. The starting point of this paperis the observation that from the conformal mapping point of view, the zero angle cusp at thetangential self-intersection for λ(t) = 4
√1 − t is very similar to the infinitely spiraling prime
end, and that this is reflected in the driving terms:
Theorem 1.1. If γ is a sufficiently smooth infinite spiral of half-plane capacity T , or if γ hasa tangential self-intersection, then its driving term λ satisfies
limt→T
|λ(T ) − λ(t)|√T − t
= 4.
Figure 1: An infinite spiral converging towards a star.
See Sections 2.1 and 5 for the definitions and precise statements. In Section 5 we show that forevery compact connected set A ⊂ H with connected complement, there is a sufficiently smoothinfinite spiral winding infinitely often around A with limit set ∂A; see Figure 1. The followingnatural question has been asked by Omer Angel: If the hull of λ is generated by a continuouscurve γ and if r < 1, is it true that the hull of rλ is generated by a continuous curve, too? Inother words, is the space of driving terms of continuous curves starlike? We answer this questionin the negative by proving
Theorem 1.2. If γ is a sufficiently smooth infinite spiral of half-plane capacity T , and if λ isits driving term, then the trace of rλ is continuous on the closed interval [0, T ] for all r = ±1.
The main work is in proving a form of stability of the (nontangential) self-intersection of λ(t) =κ√
1 − t for κ > 4 :
2
![Page 52: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/52.jpg)
Spirals and collisions
What happens when κ = 4?
4√
1− t generates a hull that intersects the real line tangentially.
The spiral behavior can be viewed as tangential behavior.
1 Introduction and Results
Let λ(t) be continuous and real valued and let gt : H \ Kt → H be the solution to the Loewnerequation
d
dtgt(z) =
2
gt(z) − λ(t), g0(z) = z ∈ H, (1.1)
where H is the upper half-plane. It was shown in [MR1] and [Li] that if λ is Holder continuouswith exponent 1/2 and if ||λ||1/2 < 4, then there is a simple curve γ with γ[0, t] = Kt and
γ \ γ(0) ⊂ H. The norm 4 is sharp as the examples λ(t) = κ√
1 − t show: Indeed, by [KNK],γ touches back on the real line if κ ≥ 4 (hence the driving term λ(t) = κ for 0 ≤ t ≤ t0 andλ(t) = κ
√t0 + 1 − t for t0 ≤ t ≤ t0 + 1 has a self-intersection in H for t0 sufficiently large).
It was also shown in [MR1] that there is a λ with ||λ||1/2 < ∞ such that K1 spirals infinitelyoften around some disc, and hence is not locally connected. The starting point of this paperis the observation that from the conformal mapping point of view, the zero angle cusp at thetangential self-intersection for λ(t) = 4
√1 − t is very similar to the infinitely spiraling prime
end, and that this is reflected in the driving terms:
Theorem 1.1. If γ is a sufficiently smooth infinite spiral of half-plane capacity T , or if γ hasa tangential self-intersection, then its driving term λ satisfies
limt→T
|λ(T ) − λ(t)|√T − t
= 4.
Figure 1: An infinite spiral converging towards a star.
See Sections 2.1 and 5 for the definitions and precise statements. In Section 5 we show that forevery compact connected set A ⊂ H with connected complement, there is a sufficiently smoothinfinite spiral winding infinitely often around A with limit set ∂A; see Figure 1. The followingnatural question has been asked by Omer Angel: If the hull of λ is generated by a continuouscurve γ and if r < 1, is it true that the hull of rλ is generated by a continuous curve, too? Inother words, is the space of driving terms of continuous curves starlike? We answer this questionin the negative by proving
Theorem 1.2. If γ is a sufficiently smooth infinite spiral of half-plane capacity T , and if λ isits driving term, then the trace of rλ is continuous on the closed interval [0, T ] for all r = ±1.
The main work is in proving a form of stability of the (nontangential) self-intersection of λ(t) =κ√
1 − t for κ > 4 :
2
![Page 53: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/53.jpg)
Recap: the Loewner equation question
L Equ
!
1
growing families of 2-d sets real-valued functions
Question: How do properties of the functions relate to geometriccharacteristics of the sets?
![Page 54: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/54.jpg)
Sample of other Loewner explorations
I Spacefilling curves (L, Rohde)
I Driving functions with higher regularity (Wong, L, Tran)
I Holder domains (Kinneberg)
I Complex-valued driving functions (Tran)
I Multiple Loewner hulls (Roth, Schleissinger, Starnes)
I Trees (Healey)
I Trace existence (Zhang, Zinsmeister)
![Page 55: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/55.jpg)
Sample of other Loewner explorations
I Spacefilling curves (L, Rohde)
I Driving functions with higher regularity (Wong, L, Tran)
I Holder domains (Kinneberg)
I Complex-valued driving functions (Tran)
I Multiple Loewner hulls (Roth, Schleissinger, Starnes)
I Trees (Healey)
I Trace existence (Zhang, Zinsmeister)
![Page 56: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/56.jpg)
Sample of other Loewner explorations
I Spacefilling curves (L, Rohde)
I Driving functions with higher regularity (Wong, L, Tran)
I Holder domains (Kinneberg)
I Complex-valued driving functions (Tran)
I Multiple Loewner hulls (Roth, Schleissinger, Starnes)
I Trees (Healey)
I Trace existence (Zhang, Zinsmeister)
![Page 57: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/57.jpg)
Sample of other Loewner explorations
I Spacefilling curves (L, Rohde)
I Driving functions with higher regularity (Wong, L, Tran)
I Holder domains (Kinneberg)
I Complex-valued driving functions (Tran)
I Multiple Loewner hulls (Roth, Schleissinger, Starnes)
I Trees (Healey)
I Trace existence (Zhang, Zinsmeister)
![Page 58: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/58.jpg)
Sample of other Loewner explorations
I Spacefilling curves (L, Rohde)
I Driving functions with higher regularity (Wong, L, Tran)
I Holder domains (Kinneberg)
I Complex-valued driving functions (Tran)
I Multiple Loewner hulls (Roth, Schleissinger, Starnes)
I Trees (Healey)
I Trace existence (Zhang, Zinsmeister)
![Page 59: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/59.jpg)
Sample of other Loewner explorations
I Spacefilling curves (L, Rohde)
I Driving functions with higher regularity (Wong, L, Tran)
I Holder domains (Kinneberg)
I Complex-valued driving functions (Tran)
I Multiple Loewner hulls (Roth, Schleissinger, Starnes)
I Trees (Healey)
I Trace existence (Zhang, Zinsmeister)
![Page 60: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/60.jpg)
Sample of other Loewner explorations
I Spacefilling curves (L, Rohde)
I Driving functions with higher regularity (Wong, L, Tran)
I Holder domains (Kinneberg)
I Complex-valued driving functions (Tran)
I Multiple Loewner hulls (Roth, Schleissinger, Starnes)
I Trees (Healey)
I Trace existence (Zhang, Zinsmeister)
![Page 61: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/61.jpg)
Sample of other Loewner explorations
I Spacefilling curves (L, Rohde)
I Driving functions with higher regularity (Wong, L, Tran)
I Holder domains (Kinneberg)
I Complex-valued driving functions (Tran)
I Multiple Loewner hulls (Roth, Schleissinger, Starnes)
I Trees (Healey)
I Trace existence (Zhang, Zinsmeister)
![Page 62: Slits, spirals and Loewner hulls =4 T 0 - Mathematicspietro/Seattle19-Slides/Joan-Lind.pdf · Joan Lind University of Tennessee. The story begins Bieberbach conjecture (1916): For](https://reader033.vdocuments.mx/reader033/viewer/2022053120/60a2ed0c2e33822130405c61/html5/thumbnails/62.jpg)
Thank you
1 Introduction and Results
Let λ(t) be continuous and real valued and let gt : H \ Kt → H be the solution to the Loewnerequation
d
dtgt(z) =
2
gt(z) − λ(t), g0(z) = z ∈ H, (1.1)
where H is the upper half-plane. It was shown in [MR1] and [Li] that if λ is Holder continuouswith exponent 1/2 and if ||λ||1/2 < 4, then there is a simple curve γ with γ[0, t] = Kt and
γ \ γ(0) ⊂ H. The norm 4 is sharp as the examples λ(t) = κ√
1 − t show: Indeed, by [KNK],γ touches back on the real line if κ ≥ 4 (hence the driving term λ(t) = κ for 0 ≤ t ≤ t0 andλ(t) = κ
√t0 + 1 − t for t0 ≤ t ≤ t0 + 1 has a self-intersection in H for t0 sufficiently large).
It was also shown in [MR1] that there is a λ with ||λ||1/2 < ∞ such that K1 spirals infinitelyoften around some disc, and hence is not locally connected. The starting point of this paperis the observation that from the conformal mapping point of view, the zero angle cusp at thetangential self-intersection for λ(t) = 4
√1 − t is very similar to the infinitely spiraling prime
end, and that this is reflected in the driving terms:
Theorem 1.1. If γ is a sufficiently smooth infinite spiral of half-plane capacity T , or if γ hasa tangential self-intersection, then its driving term λ satisfies
limt→T
|λ(T ) − λ(t)|√T − t
= 4.
Figure 1: An infinite spiral converging towards a star.
See Sections 2.1 and 5 for the definitions and precise statements. In Section 5 we show that forevery compact connected set A ⊂ H with connected complement, there is a sufficiently smoothinfinite spiral winding infinitely often around A with limit set ∂A; see Figure 1. The followingnatural question has been asked by Omer Angel: If the hull of λ is generated by a continuouscurve γ and if r < 1, is it true that the hull of rλ is generated by a continuous curve, too? Inother words, is the space of driving terms of continuous curves starlike? We answer this questionin the negative by proving
Theorem 1.2. If γ is a sufficiently smooth infinite spiral of half-plane capacity T , and if λ isits driving term, then the trace of rλ is continuous on the closed interval [0, T ] for all r = ±1.
The main work is in proving a form of stability of the (nontangential) self-intersection of λ(t) =κ√
1 − t for κ > 4 :
2