hyperbolic geometry and continued fraction theory...

Background Parabola Theorem Parabolic region Stern–Stolz series Hyperbolic geometry

Hyperbolic geometry and continued fractiontheory II

Ian Short 16 February 2010

http://maths.org/ims25/maths/presentations.php

Background

Schematic diagram

continued fractions

Mobius maps..

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Mobius group

◦ Group of hyperbolic isometries of H3.◦ Group of conformal automorphisms of C∞.

Mobius group

◦ Group of hyperbolic isometries of H3.

◦ Group of conformal automorphisms of C∞.

Mobius group

◦ Group of hyperbolic isometries of H3.◦ Group of conformal automorphisms of C∞.

Stereographic projection

The chordal metric

χ(w, z) =2|w − z|√

1 + |w|2√

1 + |z|2χ(w,∞) =

2√1 + |w|2

The chordal metric

The supremum metric

Denote the Mobius group by M.

χ0(f, g) = supz∈C∞

χ(f(z), g(z))

f, g ∈M

The metric of uniform convergence.

The supremum metric

χ(f(z), g(z))

f, g ∈M

The supremum metric

χ(f(z), g(z))

f, g ∈M

Mobius group

◦ (M, χ0) is a complete metric space◦ (M, χ0) is a topological group◦ right-invariant: χ0(fk, gk) = χ0(f, g)◦ h(z) = 1/z is a chordal isometry: χ0(hf, hg) = χ0(f, g)

Mobius group

◦ (M, χ0) is a complete metric space

◦ (M, χ0) is a topological group◦ right-invariant: χ0(fk, gk) = χ0(f, g)◦ h(z) = 1/z is a chordal isometry: χ0(hf, hg) = χ0(f, g)

Mobius group

◦ (M, χ0) is a complete metric space◦ (M, χ0) is a topological group

◦ right-invariant: χ0(fk, gk) = χ0(f, g)◦ h(z) = 1/z is a chordal isometry: χ0(hf, hg) = χ0(f, g)

Mobius group

◦ (M, χ0) is a complete metric space◦ (M, χ0) is a topological group◦ right-invariant: χ0(fk, gk) = χ0(f, g)

◦ h(z) = 1/z is a chordal isometry: χ0(hf, hg) = χ0(f, g)

Mobius group

◦ (M, χ0) is a complete metric space◦ (M, χ0) is a topological group◦ right-invariant: χ0(fk, gk) = χ0(f, g)◦ h(z) = 1/z is a chordal isometry: χ0(hf, hg) = χ0(f, g)

The Stern–Stolz Theorem

Theorem. If∑

n |bn| converges then K(1| bn) diverges.

Notation

tn(z) =1

bn + z

Tn = t1 ◦ t2 ◦ · · · ◦ tn

h(z) =1z

Notation

tn(z) =1

bn + z

Tn = t1 ◦ t2 ◦ · · · ◦ tn

h(z) =1z

Notation

tn(z) =1

bn + z

Tn = t1 ◦ t2 ◦ · · · ◦ tn

h(z) =1z

Hyperbolic geometry proof (Beardon)

Topological groups proof

χ0(Tn, Tn+2) 6∑n

T2n−1 → g T2n → gh

T2n−1(0)→ g(0) T2n(0)→ g(∞)

χ0(Tn, Tn+2) 6∑n

T2n−1 → g T2n → gh

T2n−1(0)→ g(0) T2n(0)→ g(∞)

χ0(Tn, Tn+2) 6∑n

T2n−1 → g T2n → gh

T2n−1(0)→ g(0) T2n(0)→ g(∞)

The Parabola Theorem

‘The queen of the convergence theorems’ (Lorentzen)

Topological groups techniques and hyperbolic geometry

Continued fractions

K(an| 1) =a1

1 + · · ·

Parabolic region

The Stern–Stolz series

∣∣∣∣ 1a1

∣∣∣∣+∣∣∣∣a1

∣∣∣∣+∣∣∣∣ a2

∣∣∣∣+∣∣∣∣a1a3

∣∣∣∣+∣∣∣∣ a2a4

a1a3a5

∣∣∣∣+ · · ·

Suppose that an ∈ Pα for n = 1, 2, . . . .

Then K(an| 1) convergesif and only if the series∣∣∣∣ 1

∣∣∣∣+∣∣∣∣a1

∣∣∣∣+∣∣∣∣ a2

∣∣∣∣+∣∣∣∣a1a3

∣∣∣∣+∣∣∣∣ a2a4

a1a3a5

∣∣∣∣+ · · ·

diverges.

Suppose that an ∈ Pα for n = 1, 2, . . . .Then K(an| 1) convergesif and only if the series∣∣∣∣ 1

∣∣∣∣+∣∣∣∣a1

∣∣∣∣+∣∣∣∣ a2

∣∣∣∣+∣∣∣∣a1a3

∣∣∣∣+∣∣∣∣ a2a4

a1a3a5

∣∣∣∣+ · · ·

diverges.

Long history of the Parabola Theorem

◦ Scott and Wall (Trans. Amer. Math. Soc., 1940)◦ Leighton and Thron (Duke Math. J., 1942)◦ Paydon and Wall (Duke Math. J., 1942)◦ Thron (Duke Math. J., 1943, 1944)◦ Thron (J. Indian Math. Soc., 1963)

◦ Scott and Wall (Trans. Amer. Math. Soc., 1940)

◦ Leighton and Thron (Duke Math. J., 1942)◦ Paydon and Wall (Duke Math. J., 1942)◦ Thron (Duke Math. J., 1943, 1944)◦ Thron (J. Indian Math. Soc., 1963)

◦ Scott and Wall (Trans. Amer. Math. Soc., 1940)◦ Leighton and Thron (Duke Math. J., 1942)

◦ Paydon and Wall (Duke Math. J., 1942)◦ Thron (Duke Math. J., 1943, 1944)◦ Thron (J. Indian Math. Soc., 1963)

◦ Scott and Wall (Trans. Amer. Math. Soc., 1940)◦ Leighton and Thron (Duke Math. J., 1942)◦ Paydon and Wall (Duke Math. J., 1942)

◦ Thron (Duke Math. J., 1943, 1944)◦ Thron (J. Indian Math. Soc., 1963)

◦ Scott and Wall (Trans. Amer. Math. Soc., 1940)◦ Leighton and Thron (Duke Math. J., 1942)◦ Paydon and Wall (Duke Math. J., 1942)◦ Thron (Duke Math. J., 1943, 1944)

◦ Thron (J. Indian Math. Soc., 1963)

◦ Scott and Wall (Trans. Amer. Math. Soc., 1940)◦ Leighton and Thron (Duke Math. J., 1942)◦ Paydon and Wall (Duke Math. J., 1942)◦ Thron (Duke Math. J., 1943, 1944)◦ Thron (J. Indian Math. Soc., 1963)

Understanding the theorem

What is the significance of the parabolic region?

What is the significance of the series?

Years of confusion

Split the theorem in two.

Theorem involving the parabolic region.

Theorem involving the Stern–Stolz series.

Years of confusion

Schematic diagram

Parabola Theorem

Schematic diagram

Parabola Theorem

parabolic region Stern-Stolz series

The parabolic region

Schematic diagram

Parabola Theorem

parabolic region

Mobius transformations

tn(z) =an

Tn = t1 ◦ t2 ◦ · · · ◦ tn

tn(z) =an

Tn = t1 ◦ t2 ◦ · · · ◦ tn

Convergence using Mobius transformations

The continued fraction K(an| 1) converges if and only ifT1(0), T2(0), T3(0), . . . converges.

Question

What does the condition a ∈ Pα signify for the mapt(z) = a/(1 + z)?

Answer

The coefficient a belongs to Pα if and only if t maps ahalf-plane Hα within itself.

Proof (α = 0)

t(H) ⊂ H ⇐⇒ |a− 0| 6 |a− ∂H|

Proof (α = 0)

t(H) ⊂ H ⇐⇒ |a− 0| 6 |a− ∂H|

Proof (α = 0)

t(H) ⊂ H ⇐⇒ |a− 0| 6 |a− ∂H|

Proof (α = 0)

t(H) ⊂ H ⇐⇒ |a− 0| 6 |a− ∂H|

Proof (α = 0)

t(H) ⊂ H ⇐⇒ |a− 0| 6 |a− ∂H|

Proof (α = 0)

t(H) ⊂ H ⇐⇒ |a− 0| 6 |a− ∂H|

Proof (α = 0)

t(H) ⊂ H ⇐⇒ |a− 0| 6 |a− ∂H|

Proof (α = 0)

Parabola |a− 0| = |a− ∂H|

Original condition

an ∈ Pα

New condition

tn(−1) =∞

tn(∞) = 0 tn(Hα) ⊂ Hα

Does Tn = t1 ◦ t2 ◦ · · · ◦ tn converge at 0?

New condition

tn(−1) =∞ tn(∞) = 0

tn(Hα) ⊂ Hα

New condition

tn(−1) =∞ tn(∞) = 0 tn(Hα) ⊂ Hα

New condition

tn(−1) =∞ tn(∞) = 0 tn(Hα) ⊂ Hα

Extensive literature

◦ Hillam and Thron (Proc. Amer. Math. Soc., 1965)◦ Baker and Rippon (Complex Variables, 1989)◦ Baker and Rippon (Arch. Math., 1990)◦ Beardon (Comp. Methods. Func. Theory, 2001)◦ Beardon, Carne, Minda, and Ng (Ergod. Th. & Dynam.

Sys., 2004)◦ Lorentzen (Ramanujan J., 2007)

◦ Hillam and Thron (Proc. Amer. Math. Soc., 1965)

◦ Baker and Rippon (Complex Variables, 1989)◦ Baker and Rippon (Arch. Math., 1990)◦ Beardon (Comp. Methods. Func. Theory, 2001)◦ Beardon, Carne, Minda, and Ng (Ergod. Th. & Dynam.

◦ Hillam and Thron (Proc. Amer. Math. Soc., 1965)◦ Baker and Rippon (Complex Variables, 1989)

◦ Baker and Rippon (Arch. Math., 1990)◦ Beardon (Comp. Methods. Func. Theory, 2001)◦ Beardon, Carne, Minda, and Ng (Ergod. Th. & Dynam.

◦ Hillam and Thron (Proc. Amer. Math. Soc., 1965)◦ Baker and Rippon (Complex Variables, 1989)◦ Baker and Rippon (Arch. Math., 1990)

◦ Beardon (Comp. Methods. Func. Theory, 2001)◦ Beardon, Carne, Minda, and Ng (Ergod. Th. & Dynam.

◦ Hillam and Thron (Proc. Amer. Math. Soc., 1965)◦ Baker and Rippon (Complex Variables, 1989)◦ Baker and Rippon (Arch. Math., 1990)◦ Beardon (Comp. Methods. Func. Theory, 2001)

◦ Beardon, Carne, Minda, and Ng (Ergod. Th. & Dynam.Sys., 2004)◦ Lorentzen (Ramanujan J., 2007)

Sys., 2004)

◦ Lorentzen (Ramanujan J., 2007)

Conclusion

If an ∈ Pα then there are points p and q in Hα such that T2n−1

converges on Hα to p, and T2n converges on Hα to q.

Divergence

Action of T2n−1

Divergence

Action of T2n

Summary

◦ an ∈ Pα◦ tn(Hα) ⊆ Hα

◦ refer to the literature◦ T2n−1 → p and T2n → q

Summary

◦ an ∈ Pα

◦ tn(Hα) ⊆ Hα

Summary

◦ refer to the literature

◦ T2n−1 → p and T2n → q

Summary

Schematic diagram

Parabola Theorem

Stern-Stolz series

Recall the Parabola Theorem

Suppose an ∈ Pα.

Then K(an| 1) converges if and only if theStern–Stolz series diverges.

Suppose an ∈ Pα. Then K(an| 1) converges if and only if theStern–Stolz series diverges.

Suppose an ∈ Pα.

Then K(an| 1) converges if and only if theStern–Stolz series diverges.

∣∣∣∣ 1a1

∣∣∣∣+∣∣∣∣a1

∣∣∣∣+∣∣∣∣ a2

∣∣∣∣+∣∣∣∣a1a3

∣∣∣∣+∣∣∣∣ a2a4

a1a3a5

∣∣∣∣+ · · ·

Convergence of the Stern–Stolz series

tn(z) =an

1 + z∼ sn(z) =

sn(z) =anz

Sn = s1 ◦ s2 ◦ · · · ◦ sn

sn(z) =anz

Sn = s1 ◦ s2 ◦ · · · ◦ sn

Is Sn ∼ Tn?

Recall supremum metric

◦ χ chordal metric◦ M Mobius group◦ χ0(f, g) = supz∈C∞ χ(f(z), g(z))◦ χ0 right-invariant◦ (M, χ0) a topological group◦ (M, χ0) a complete metric space

◦ χ chordal metric

◦ M Mobius group◦ χ0(f, g) = supz∈C∞ χ(f(z), g(z))◦ χ0 right-invariant◦ (M, χ0) a topological group◦ (M, χ0) a complete metric space

◦ χ chordal metric◦ M Mobius group

◦ χ0(f, g) = supz∈C∞ χ(f(z), g(z))◦ χ0 right-invariant◦ (M, χ0) a topological group◦ (M, χ0) a complete metric space

◦ χ chordal metric◦ M Mobius group◦ χ0(f, g) = supz∈C∞ χ(f(z), g(z))

◦ χ0 right-invariant◦ (M, χ0) a topological group◦ (M, χ0) a complete metric space

◦ χ chordal metric◦ M Mobius group◦ χ0(f, g) = supz∈C∞ χ(f(z), g(z))◦ χ0 right-invariant

◦ (M, χ0) a topological group◦ (M, χ0) a complete metric space

◦ χ chordal metric◦ M Mobius group◦ χ0(f, g) = supz∈C∞ χ(f(z), g(z))◦ χ0 right-invariant◦ (M, χ0) a topological group

◦ (M, χ0) a complete metric space

◦ χ chordal metric◦ M Mobius group◦ χ0(f, g) = supz∈C∞ χ(f(z), g(z))◦ χ0 right-invariant◦ (M, χ0) a topological group◦ (M, χ0) a complete metric space

µ1 =1a1

µ2 =a1

a2µ3 =

a1a3. . .

|µ1|+ |µ2|+ |µ3|+ · · ·

S2n−1(z) =1

µ2n−1zS2n(z) = µ2nz

µ1 =1a1

µ2 =a1

a2µ3 =

a1a3. . .

|µ1|+ |µ2|+ |µ3|+ · · ·

S2n−1(z) =1

µ1 =1a1

µ2 =a1

a2µ3 =

a1a3. . .

|µ1|+ |µ2|+ |µ3|+ · · ·

S2n−1(z) =1

Conjugation

µz ◦ (1 + z) ◦ µ−1z = µ+ z

S2n ◦ (1 + z) ◦ S−12n (z) = µ2n + z

h(z) =1z

S2n−1 ◦ (1 + z) ◦ S−12n−1(z) = h ◦ (µ2n−1 + z) ◦ h

Conjugation

µz ◦ (1 + z) ◦ µ−1z = µ+ z

S2n ◦ (1 + z) ◦ S−12n (z)

= µ2n + z

h(z) =1z

S2n−1 ◦ (1 + z) ◦ S−12n−1(z) = h ◦ (µ2n−1 + z) ◦ h

Conjugation

µz ◦ (1 + z) ◦ µ−1z = µ+ z

S2n ◦ (1 + z) ◦ S−12n (z) = µ2n + z

h(z) =1z

S2n−1 ◦ (1 + z) ◦ S−12n−1(z) = h ◦ (µ2n−1 + z) ◦ h

Conjugation

µz ◦ (1 + z) ◦ µ−1z = µ+ z

S2n ◦ (1 + z) ◦ S−12n (z) = µ2n + z

h(z) =1z

S2n−1 ◦ (1 + z) ◦ S−12n−1(z) = h ◦ (µ2n−1 + z) ◦ h

Conjugation

µz ◦ (1 + z) ◦ µ−1z = µ+ z

S2n ◦ (1 + z) ◦ S−12n (z) = µ2n + z

h(z) =1z

S2n−1 ◦ (1 + z) ◦ S−12n−1(z)

= h ◦ (µ2n−1 + z) ◦ h

Conjugation

µz ◦ (1 + z) ◦ µ−1z = µ+ z

S2n ◦ (1 + z) ◦ S−12n (z) = µ2n + z

h(z) =1z

S2n−1 ◦ (1 + z) ◦ S−12n−1(z) = h ◦ (µ2n−1 + z) ◦ h

Calculation

χ0(SnT−1n , Sn−1T

−1n−1)

= χ0(Snt−1n , Sn−1)

= χ0(I, Sn−1tnS−1n )

= χ0(I, Sn ◦ (1 + z) ◦ S−1n )

= χ0(I, µn + z)∼ |µn|

Calculation

−1n−1) = χ0(Snt−1

n , Sn−1)

= χ0(I, Sn ◦ (1 + z) ◦ S−1n )

= χ0(I, µn + z)∼ |µn|

Calculation

−1n−1) = χ0(Snt−1

n , Sn−1)

= χ0(I, Sn ◦ (1 + z) ◦ S−1n )

= χ0(I, µn + z)∼ |µn|

Calculation

−1n−1) = χ0(Snt−1

n , Sn−1)

= χ0(I, Sn ◦ (1 + z) ◦ S−1n )

= χ0(I, µn + z)∼ |µn|

Calculation

−1n−1) = χ0(Snt−1

n , Sn−1)

= χ0(I, Sn ◦ (1 + z) ◦ S−1n )

= χ0(I, µn + z)

∼ |µn|

Calculation

−1n−1) = χ0(Snt−1

n , Sn−1)

= χ0(I, Sn ◦ (1 + z) ◦ S−1n )

= χ0(I, µn + z)∼ |µn|

Summary

∣∣∣∣ 1a1

∣∣∣∣+∣∣∣∣a1

∣∣∣∣+∣∣∣∣ a2

∣∣∣∣+∣∣∣∣a1a3

∣∣∣∣+∣∣∣∣ a2a4

a1a3a5

∣∣∣∣+ · · · < +∞

if and only if∑n

−1n−1) < +∞

There exists Mobius f such that χ0(SnT−1n , f)→ 0.

Let g = f−1.

χ0(gSn, Tn)→ 0

−1n−1) < +∞

Let g = f−1.

χ0(gSn, Tn)→ 0

−1n−1) < +∞

Let g = f−1.

χ0(gSn, Tn)→ 0

−1n−1) < +∞

Let g = f−1.

χ0(gSn, Tn)→ 0

−1n−1) < +∞

Let g = f−1.

χ0(gSn, Tn)→ 0

Oscillation

S2n−1(z) =1

µn → 0

S2n−1 →∞ S2n → 0

So if Tn ∼ gSn then T2n−1 → g(∞) and T2n → g(0).

Oscillation

S2n−1(z) =1

µn → 0

S2n−1 →∞ S2n → 0

Oscillation

S2n−1(z) =1

µn → 0

S2n−1 →∞ S2n → 0

Oscillation

S2n−1(z) =1

µn → 0

S2n−1 →∞ S2n → 0

Oscillation

Action of T2n−1

Oscillation

Action of T2n

Open Problem III

What is the significance, if any, of the many other versions ofthe Parabola Theorem? (See earlier references and Lorentzenand Waadeland book.)

Open Problem III

What is the significance, if any, of the many other versions ofthe Parabola Theorem?

(See earlier references and Lorentzenand Waadeland book.)

Open Problem III

What is the significance, if any, of the many other versions ofthe Parabola Theorem? (See earlier references and Lorentzenand Waadeland book.)

Hyperbolic geometry

Hyperbolic space

Hyperbolic geometry

S2n−1(z) =1

S−12n−1(z) =

1µ2n−1z

S−12n (z) =

S−1n (j) =

If |µn| < 1 then

exp[−ρ(j, S−1

n (j))]

= exp[− log

(1|µn|

)]= |µn|

Hyperbolic geometry

S2n−1(z) =1

S−12n−1(z) =

1µ2n−1z

S−12n (z) =

S−1n (j) =

If |µn| < 1 then

exp[−ρ(j, S−1

n (j))]

= exp[− log

(1|µn|

)]= |µn|

Hyperbolic geometry

S2n−1(z) =1

S−12n−1(z) =

1µ2n−1z

S−12n (z) =

S−1n (j) =

If |µn| < 1 then

exp[−ρ(j, S−1

n (j))]

= exp[− log

(1|µn|

)]= |µn|

Hyperbolic geometry

S2n−1(z) =1

S−12n−1(z) =

1µ2n−1z

S−12n (z) =

S−1n (j) =

If |µn| < 1 then

exp[−ρ(j, S−1

n (j))]

= exp[− log

(1|µn|

)]= |µn|

Hyperbolic geometry

S2n−1(z) =1

S−12n−1(z) =

1µ2n−1z

S−12n (z) =

S−1n (j) =

If |µn| < 1 then

exp[−ρ(j, S−1

n (j))]

= exp[− log

(1|µn|

= |µn|

Hyperbolic geometry

S2n−1(z) =1

S−12n−1(z) =

1µ2n−1z

S−12n (z) =

S−1n (j) =

If |µn| < 1 then

exp[−ρ(j, S−1

n (j))]

= exp[− log

(1|µn|

)]= |µn|

Dynamics of Sn in hyperbolic space

Equivalent conditions

∣∣∣∣ 1a1

∣∣∣∣+∣∣∣∣a1

∣∣∣∣+∣∣∣∣ a2

∣∣∣∣+∣∣∣∣a1a3

∣∣∣∣+∣∣∣∣ a2a4

a1a3a5

∣∣∣∣+ · · · < +∞

−1n−1) < +∞

∑n exp[−ρ(j, Sn(j))] < +∞ and ∞ is the only (conical) limit

point of Sn∑n exp[−ρ(j, Tn(j))] < +∞ and ∞ is the only conical limit

point of Tn

∣∣∣∣ 1a1

∣∣∣∣+∣∣∣∣a1

∣∣∣∣+∣∣∣∣ a2

∣∣∣∣+∣∣∣∣a1a3

∣∣∣∣+∣∣∣∣ a2a4

a1a3a5

∣∣∣∣+ · · · < +∞

−1n−1) < +∞

point of Tn

∣∣∣∣ 1a1

∣∣∣∣+∣∣∣∣a1

∣∣∣∣+∣∣∣∣ a2

∣∣∣∣+∣∣∣∣a1a3

∣∣∣∣+∣∣∣∣ a2a4

a1a3a5

∣∣∣∣+ · · · < +∞

−1n−1) < +∞

point of Sn

∑n exp[−ρ(j, Tn(j))] < +∞ and ∞ is the only conical limit

point of Tn

∣∣∣∣ 1a1

∣∣∣∣+∣∣∣∣a1

∣∣∣∣+∣∣∣∣ a2

∣∣∣∣+∣∣∣∣a1a3

∣∣∣∣+∣∣∣∣ a2a4

a1a3a5

∣∣∣∣+ · · · < +∞

−1n−1) < +∞

point of Tn

hyperbolic geometry and continued fraction theory...

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