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$: Hyperbolic geometry and continued fraction theory IIusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010b.pdf · Background Parabola Theorem Parabolic region Stern{Stolz series$
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.


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


Hyperbolic geometry proof (Beardon)


Topological groups proof

∑n

χ0(Tn, Tn+2) 6∑n

|bn|

T2n−1 → g T2n → gh

T2n−1(0)→ g(0) T2n(0)→ g(∞)


Today

The Parabola Theorem

‘The queen of the convergence theorems’ (Lorentzen)

Topological groups techniques and hyperbolic geometry


Continued fractions

K(an| 1) =a1

1 +a2

1 +a3

1 +a4

1 + · · ·


Parabolic region


The Stern–Stolz series

∣∣∣∣ 1a1

∣∣∣∣+∣∣∣∣a1

a2

∣∣∣∣+∣∣∣∣ a2

a1a3

∣∣∣∣+∣∣∣∣a1a3

a2a4

∣∣∣∣+∣∣∣∣ a2a4

a1a3a5

∣∣∣∣+ · · ·



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

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

a1

∣∣∣∣+∣∣∣∣a1

a2

∣∣∣∣+∣∣∣∣ a2

a1a3

∣∣∣∣+∣∣∣∣a1a3

a2a4

∣∣∣∣+∣∣∣∣ a2a4

a1a3a5

∣∣∣∣+ · · ·

diverges.



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

a1

∣∣∣∣+∣∣∣∣a1

a2

∣∣∣∣+∣∣∣∣ a2

a1a3

∣∣∣∣+∣∣∣∣a1a3

a2a4

∣∣∣∣+∣∣∣∣ 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.


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

1 + z

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)

Parabola |a− 0| = |a− ∂H|


Original condition

an ∈ Pα


New condition


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



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




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

∣∣∣∣+∣∣∣∣ a2

a1a3

∣∣∣∣+∣∣∣∣a1a3

a2a4

∣∣∣∣+∣∣∣∣ a2a4

a1a3a5

∣∣∣∣+ · · ·


Convergence of the Stern–Stolz series

tn(z) =an

1 + z∼ sn(z) =

anz


Mobius transformations

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 =

a2

a1a3. . .

|µ1|+ |µ2|+ |µ3|+ · · ·

S2n−1(z) =1

µ2n−1zS2n(z) = µ2nz


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|


Summary

∣∣∣∣ 1a1

∣∣∣∣+∣∣∣∣a1

a2

∣∣∣∣+∣∣∣∣ a2

a1a3

∣∣∣∣+∣∣∣∣a1a3

a2a4

∣∣∣∣+∣∣∣∣ a2a4

a1a3a5

∣∣∣∣+ · · · < +∞

if and only if∑n


−1n−1) < +∞



∑n


−1n−1) < +∞

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

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

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) =

z

µ2n

S−1n (j) =

j

|µn|

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) =

z

µ2n

S−1n (j) =

j

|µn|

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) =

z

µ2n

S−1n (j) =

j

|µn|

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

∣∣∣∣+∣∣∣∣ a2

a1a3

∣∣∣∣+∣∣∣∣a1a3

a2a4

∣∣∣∣+∣∣∣∣ a2a4

a1a3a5

∣∣∣∣+ · · · < +∞

∑n


−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

∣∣∣∣+∣∣∣∣ a2

a1a3

∣∣∣∣+∣∣∣∣a1a3

a2a4

∣∣∣∣+∣∣∣∣ a2a4

a1a3a5

∣∣∣∣+ · · · < +∞

∑n


−1n−1) < +∞


point of Sn

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

point of Tn



∣∣∣∣ 1a1

∣∣∣∣+∣∣∣∣a1

a2

∣∣∣∣+∣∣∣∣ a2

a1a3

∣∣∣∣+∣∣∣∣a1a3

a2a4

∣∣∣∣+∣∣∣∣ a2a4

a1a3a5

∣∣∣∣+ · · · < +∞

∑n


−1n−1) < +∞


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

point of Tn


1

T +1

H +1

A +1

N +1

K +1

Y +1

O +1

U +1!

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