A classical theorem The parabolic region The Stern–Stolz series

The Parabola Theorem on continued fractions

Ian Short

7 September 2009

A classical theorem The parabolic region The Stern–Stolz series

A classical theorem

A classical theorem The parabolic region The Stern–Stolz series

Continued fractions

K(an| 1) =a1

1 +a2

1 +a3

1 +a4

1 + · · ·

A classical theorem The parabolic region The Stern–Stolz series

Continued fraction convergence

a11,

a1

1 +a2

1

,a1

1 +a2

1 +a3

1

, . . .

A classical theorem The parabolic region The Stern–Stolz series

Parabolic region

A classical theorem The parabolic region The Stern–Stolz series

The Parabola Theorem

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

Then K(an| 1) convergesif and only if the series∣∣∣∣ 1a1

∣∣∣∣+ ∣∣∣∣a1a2∣∣∣∣+ ∣∣∣∣ a2a1a3

∣∣∣∣+ ∣∣∣∣a1a3a2a4∣∣∣∣+ ∣∣∣∣ a2a4a1a3a5

∣∣∣∣+ · · ·diverges.

A classical theorem The parabolic region The Stern–Stolz series

The Parabola Theorem

Suppose that an ∈ Pα for n = 1, 2, . . . .Then K(an| 1) convergesif and only if the series∣∣∣∣ 1a1

∣∣∣∣+ ∣∣∣∣a1a2∣∣∣∣+ ∣∣∣∣ a2a1a3

∣∣∣∣+ ∣∣∣∣a1a3a2a4∣∣∣∣+ ∣∣∣∣ a2a4a1a3a5

∣∣∣∣+ · · ·diverges.

A classical theorem The parabolic region The Stern–Stolz series

Understanding the theorem

What is the significance of the parabolic region?

What is the signifance of the series?

A classical theorem The parabolic region The Stern–Stolz series

Understanding the theorem

What is the significance of the parabolic region?

What is the signifance of the series?

A classical theorem The parabolic region The Stern–Stolz series

Understanding the theorem

What is the significance of the parabolic region?

What is the signifance of the series?

A classical theorem The parabolic region The Stern–Stolz series

Years of confusion

Split the theorem in two.

Theorem involving the parabolic region.

Theorem involving the Stern–Stolz series.

A classical theorem The parabolic region The Stern–Stolz series

Years of confusion

Split the theorem in two.

Theorem involving the parabolic region.

Theorem involving the Stern–Stolz series.

A classical theorem The parabolic region The Stern–Stolz series

Years of confusion

Split the theorem in two.

Theorem involving the parabolic region.

Theorem involving the Stern–Stolz series.

A classical theorem The parabolic region The Stern–Stolz series

Years of confusion

Split the theorem in two.

Theorem involving the parabolic region.

Theorem involving the Stern–Stolz series.

A classical theorem The parabolic region The Stern–Stolz series

The parabolic region

A classical theorem The parabolic region The Stern–Stolz series

Möbius transformations

tn(z) =an

1 + z

Tn = t1 ◦ t2 ◦ · · · ◦ tn

A classical theorem The parabolic region The Stern–Stolz series

Möbius transformations

tn(z) =an

1 + z

Tn = t1 ◦ t2 ◦ · · · ◦ tn

A classical theorem The parabolic region The Stern–Stolz series

Convergence using Möbius transformations

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

A classical theorem The parabolic region The Stern–Stolz series

Question

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

A classical theorem The parabolic region The Stern–Stolz series

Answer

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

A classical theorem The parabolic region The Stern–Stolz series

Proof (α = 0)

A classical theorem The parabolic region The Stern–Stolz series

Proof (α = 0)

A classical theorem The parabolic region The Stern–Stolz series

Proof (α = 0)

A classical theorem The parabolic region The Stern–Stolz series

Proof (α = 0)

A classical theorem The parabolic region The Stern–Stolz series

Proof (α = 0)

A classical theorem The parabolic region The Stern–Stolz series

Proof (α = 0)

t(H) ⊂ H ⇐⇒ |a− 0| ≤ |a− ∂H|

A classical theorem The parabolic region The Stern–Stolz series

Proof (α = 0)

Parabola |a− 0| = |a− ∂H|.

A classical theorem The parabolic region The Stern–Stolz series

Original condition

an ∈ Pα

A classical theorem The parabolic region The Stern–Stolz series

New condition

A classical theorem The parabolic region The Stern–Stolz series

New condition

tn(−1) =∞

tn(∞) = 0 tn(Hα) ⊂ Hα

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

A classical theorem The parabolic region The Stern–Stolz series

New condition

tn(−1) =∞ tn(∞) = 0

tn(Hα) ⊂ Hα

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

A classical theorem The parabolic region The Stern–Stolz series

New condition

tn(−1) =∞ tn(∞) = 0 tn(Hα) ⊂ Hα

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

A classical theorem The parabolic region The Stern–Stolz series

New condition

tn(−1) =∞ tn(∞) = 0 tn(Hα) ⊂ Hα

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

A classical theorem The parabolic region The Stern–Stolz series

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)

A classical theorem The parabolic region The Stern–Stolz series

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)

A classical theorem The parabolic region The Stern–Stolz series

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)

A classical theorem The parabolic region The Stern–Stolz series

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)

A classical theorem The parabolic region The Stern–Stolz series

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)

A classical theorem The parabolic region The Stern–Stolz series

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)

A classical theorem The parabolic region The Stern–Stolz series

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)

A classical theorem The parabolic region The Stern–Stolz series

Conclusion

If an ∈ Pα then there are points p and q in Hα such that T2n−1converges on Hα to p, and T2n converges on Hα to q.

A classical theorem The parabolic region The Stern–Stolz series

Divergence

Action of T2n−1

A classical theorem The parabolic region The Stern–Stolz series

Divergence

Action of T2n

A classical theorem The parabolic region The Stern–Stolz series

The Stern–Stolz series

A classical theorem The parabolic region The Stern–Stolz series

Recall

Suppose an ∈ Pα.

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

A classical theorem The parabolic region The Stern–Stolz series

Recall

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

A classical theorem The parabolic region The Stern–Stolz series

Recall

Suppose an ∈ Pα.

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

A classical theorem The parabolic region The Stern–Stolz series

The Stern–Stolz series

∣∣∣∣ 1a1∣∣∣∣+ ∣∣∣∣a1a2

∣∣∣∣+ ∣∣∣∣ a2a1a3∣∣∣∣+ ∣∣∣∣a1a3a2a4

∣∣∣∣+ ∣∣∣∣ a2a4a1a3a5∣∣∣∣+ · · ·

A classical theorem The parabolic region The Stern–Stolz series

Convergence of the Stern–Stolz series

tn(z) =an

1 + z∼ sn(z) =

anz

A classical theorem The parabolic region The Stern–Stolz series

Möbius transformations

sn(z) =anz

Sn = s1 ◦ s2 ◦ · · · ◦ sn

A classical theorem The parabolic region The Stern–Stolz series

Möbius transformations

sn(z) =anz

Sn = s1 ◦ s2 ◦ · · · ◦ sn

A classical theorem The parabolic region The Stern–Stolz series

Convergence of the Stern–Stolz series

Is Sn ∼ Tn?

A classical theorem The parabolic region The Stern–Stolz series

Chordal metric

Let χ denote the chordal metric.

Let M denote the group of Möbius transformations.

χ0(f, g) = supz∈C∞

χ(f(z), g(z))

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

A classical theorem The parabolic region The Stern–Stolz series

Chordal metric

Let χ denote the chordal metric.Let M denote the group of Möbius transformations.

χ0(f, g) = supz∈C∞

χ(f(z), g(z))

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

A classical theorem The parabolic region The Stern–Stolz series

Chordal metric

Let χ denote the chordal metric.Let M denote the group of Möbius transformations.

χ0(f, g) = supz∈C∞

χ(f(z), g(z))

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

A classical theorem The parabolic region The Stern–Stolz series

Chordal metric

Let χ denote the chordal metric.Let M denote the group of Möbius transformations.

χ0(f, g) = supz∈C∞

χ(f(z), g(z))

χ0 is right-invariant.

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

A classical theorem The parabolic region The Stern–Stolz series

Chordal metric

Let χ denote the chordal metric.Let M denote the group of Möbius transformations.

χ0(f, g) = supz∈C∞

χ(f(z), g(z))

χ0 is right-invariant.(M, χ0) is a topological group.

(M, χ0) is a complete metric space.

A classical theorem The parabolic region The Stern–Stolz series

Chordal metric

Let χ denote the chordal metric.Let M denote the group of Möbius transformations.

χ0(f, g) = supz∈C∞

χ(f(z), g(z))

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

A classical theorem The parabolic region The Stern–Stolz series

The Stern–Stolz series

µ1 =1a1

µ2 =a2a1

µ3 =a2a1a3

. . .

|µ1|+ |µ2|+ |µ3|+ · · ·

S2n−1(z) =1

µ2n−1zS2n(z) = µ2nz

A classical theorem The parabolic region The Stern–Stolz series

The Stern–Stolz series

µ1 =1a1

µ2 =a2a1

µ3 =a2a1a3

. . .

|µ1|+ |µ2|+ |µ3|+ · · ·

S2n−1(z) =1

µ2n−1zS2n(z) = µ2nz

A classical theorem The parabolic region The Stern–Stolz series

The Stern–Stolz series

µ1 =1a1

µ2 =a2a1

µ3 =a2a1a3

. . .

|µ1|+ |µ2|+ |µ3|+ · · ·

S2n−1(z) =1

µ2n−1zS2n(z) = µ2nz

A classical theorem The parabolic region The Stern–Stolz series

Conjugation

µz ◦ (1 + z) ◦ µ−1z = µ+ z

S2n ◦ (1 + z) ◦ S−12n (z) = µ2n + z

Let σ(z) = 1z .

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

A classical theorem The parabolic region The Stern–Stolz series

Conjugation

µz ◦ (1 + z) ◦ µ−1z = µ+ z

S2n ◦ (1 + z) ◦ S−12n (z)

= µ2n + z

Let σ(z) = 1z .

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

A classical theorem The parabolic region The Stern–Stolz series

Conjugation

µz ◦ (1 + z) ◦ µ−1z = µ+ z

S2n ◦ (1 + z) ◦ S−12n (z) = µ2n + z

Let σ(z) = 1z .

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

A classical theorem The parabolic region The Stern–Stolz series

Conjugation

µz ◦ (1 + z) ◦ µ−1z = µ+ z

S2n ◦ (1 + z) ◦ S−12n (z) = µ2n + z

Let σ(z) = 1z .

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

A classical theorem The parabolic region The Stern–Stolz series

Conjugation

µz ◦ (1 + z) ◦ µ−1z = µ+ z

S2n ◦ (1 + z) ◦ S−12n (z) = µ2n + z

Let σ(z) = 1z .

S2n−1 ◦ (1 + z) ◦ S−12n−1(z)

= σ ◦ (µ2n−1 + z) ◦ σ

A classical theorem The parabolic region The Stern–Stolz series

Conjugation

µz ◦ (1 + z) ◦ µ−1z = µ+ z

S2n ◦ (1 + z) ◦ S−12n (z) = µ2n + z

Let σ(z) = 1z .

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

A classical theorem The parabolic region The Stern–Stolz series

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|

A classical theorem The parabolic region The Stern–Stolz series

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|

A classical theorem The parabolic region The Stern–Stolz series

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|

A classical theorem The parabolic region The Stern–Stolz series

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|

A classical theorem The parabolic region The Stern–Stolz series

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|

A classical theorem The parabolic region The Stern–Stolz series

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|

A classical theorem The parabolic region The Stern–Stolz series

Summary

∣∣∣∣ 1a1∣∣∣∣+ ∣∣∣∣a1a2

∣∣∣∣+ ∣∣∣∣ a2a1a3∣∣∣∣+ ∣∣∣∣a1a3a2a4

∣∣∣∣+ ∣∣∣∣ a2a4a1a3a5∣∣∣∣+ · · · < +∞

if and only if∑n

χ0(SnT−1n , Sn−1T−1n−1) < +∞

A classical theorem The parabolic region The Stern–Stolz series

Convergence of the Stern–Stolz series

∑n

χ0(SnT−1n , Sn−1T−1n−1) < +∞

There exists Möbius g such that χ0(SnT−1n , g)→ 0.

Let h = g−1.

χ0(Tn, hSn)→ 0

A classical theorem The parabolic region The Stern–Stolz series

Convergence of the Stern–Stolz series

∑n

χ0(SnT−1n , Sn−1T−1n−1) < +∞

There exists Möbius g such that χ0(SnT−1n , g)→ 0.

Let h = g−1.

χ0(Tn, hSn)→ 0

A classical theorem The parabolic region The Stern–Stolz series

Convergence of the Stern–Stolz series

∑n

χ0(SnT−1n , Sn−1T−1n−1) < +∞

There exists Möbius g such that χ0(SnT−1n , g)→ 0.

Let h = g−1.

χ0(Tn, hSn)→ 0

A classical theorem The parabolic region The Stern–Stolz series

Convergence of the Stern–Stolz series

∑n

χ0(SnT−1n , Sn−1T−1n−1) < +∞

There exists Möbius g such that χ0(SnT−1n , g)→ 0.

Let h = g−1.

χ0(Tn, hSn)→ 0

A classical theorem The parabolic region The Stern–Stolz series

Convergence of the Stern–Stolz series

∑n

χ0(SnT−1n , Sn−1T−1n−1) < +∞

There exists Möbius g such that χ0(SnT−1n , g)→ 0.

Let h = g−1.

χ0(Tn, hSn)→ 0

A classical theorem The parabolic region The Stern–Stolz series

Oscillation

RecallS2n−1(z) =

1µ2n−1z

S2n(z) = µ2nz

So if Tn ∼ hSn then T2n−1 → h(∞) and T2n → h(0).

A classical theorem The parabolic region The Stern–Stolz series

Oscillation

RecallS2n−1(z) =

1µ2n−1z

S2n(z) = µ2nz

So if Tn ∼ hSn then T2n−1 → h(∞) and T2n → h(0).

A classical theorem The parabolic region The Stern–Stolz series

Oscillation

Action of T2n−1

A classical theorem The parabolic region The Stern–Stolz series

Oscillation

Action of T2n

A classical theorem The parabolic region The Stern–Stolz series

Hyperbolic space

A classical theorem The parabolic region The Stern–Stolz series

Hyperbolic geometry

S2n−1(z) =1

µ2n−1zS2n(z) = µ2nz

S−12n−1(z) =1

µ2n−1zS−12n (z) =

z

µ2n

S−1n (j) =j

|µn|

If |µn| < 1 then

exp[−ρ(j, S−1n (j))

]= exp

[− log

(1|µn|

)]= |µn|

A classical theorem The parabolic region The Stern–Stolz series

Hyperbolic geometry

S2n−1(z) =1

µ2n−1zS2n(z) = µ2nz

S−12n−1(z) =1

µ2n−1zS−12n (z) =

z

µ2n

S−1n (j) =j

|µn|

If |µn| < 1 then

exp[−ρ(j, S−1n (j))

]= exp

[− log

(1|µn|

)]= |µn|

A classical theorem The parabolic region The Stern–Stolz series

Hyperbolic geometry

S2n−1(z) =1

µ2n−1zS2n(z) = µ2nz

S−12n−1(z) =1

µ2n−1zS−12n (z) =

z

µ2n

S−1n (j) =j

|µn|

If |µn| < 1 then

exp[−ρ(j, S−1n (j))

]= exp

[− log

(1|µn|

)]= |µn|

A classical theorem The parabolic region The Stern–Stolz series

Hyperbolic geometry

S2n−1(z) =1

µ2n−1zS2n(z) = µ2nz

S−12n−1(z) =1

µ2n−1zS−12n (z) =

z

µ2n

S−1n (j) =j

|µn|

If |µn| < 1 then

exp[−ρ(j, S−1n (j))

]

= exp[− log

(1|µn|

)]= |µn|

A classical theorem The parabolic region The Stern–Stolz series

Hyperbolic geometry

S2n−1(z) =1

µ2n−1zS2n(z) = µ2nz

S−12n−1(z) =1

µ2n−1zS−12n (z) =

z

µ2n

S−1n (j) =j

|µn|

If |µn| < 1 then

exp[−ρ(j, S−1n (j))

]= exp

[− log

(1|µn|

)]

= |µn|

A classical theorem The parabolic region The Stern–Stolz series

Hyperbolic geometry

S2n−1(z) =1

µ2n−1zS2n(z) = µ2nz

S−12n−1(z) =1

µ2n−1zS−12n (z) =

z

µ2n

S−1n (j) =j

|µn|

If |µn| < 1 then

exp[−ρ(j, S−1n (j))

]= exp

[− log

(1|µn|

)]= |µn|

A classical theorem The parabolic region The Stern–Stolz series

Dynamics of Sn in hyperbolic space

A classical theorem The parabolic region The Stern–Stolz series

Dynamics of Sn in hyperbolic space

A classical theorem The parabolic region The Stern–Stolz series

Dynamics of Sn in hyperbolic space

A classical theorem The parabolic region The Stern–Stolz series

Dynamics of Sn in hyperbolic space

A classical theorem The parabolic region The Stern–Stolz series

Dynamics of Sn in hyperbolic space

A classical theorem The parabolic region The Stern–Stolz series

Dynamics of Sn in hyperbolic space

A classical theorem The parabolic region The Stern–Stolz series

Equivalent conditions

∣∣∣∣ 1a1∣∣∣∣+ ∣∣∣∣a1a2

∣∣∣∣+ ∣∣∣∣ a2a1a3∣∣∣∣+ ∣∣∣∣a1a3a2a4

∣∣∣∣+ ∣∣∣∣ a2a4a1a3a5∣∣∣∣+ · · · < +∞

∑n

χ0(SnT−1n , Sn−1T−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

A classical theorem The parabolic region The Stern–Stolz series

Equivalent conditions

∣∣∣∣ 1a1∣∣∣∣+ ∣∣∣∣a1a2

∣∣∣∣+ ∣∣∣∣ a2a1a3∣∣∣∣+ ∣∣∣∣a1a3a2a4

∣∣∣∣+ ∣∣∣∣ a2a4a1a3a5∣∣∣∣+ · · · < +∞∑

n

χ0(SnT−1n , Sn−1T−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

A classical theorem The parabolic region The Stern–Stolz series

Equivalent conditions

∣∣∣∣ 1a1∣∣∣∣+ ∣∣∣∣a1a2

∣∣∣∣+ ∣∣∣∣ a2a1a3∣∣∣∣+ ∣∣∣∣a1a3a2a4

∣∣∣∣+ ∣∣∣∣ a2a4a1a3a5∣∣∣∣+ · · · < +∞∑

n

χ0(SnT−1n , Sn−1T−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

A classical theorem The parabolic region The Stern–Stolz series

Equivalent conditions

∣∣∣∣ 1a1∣∣∣∣+ ∣∣∣∣a1a2

∣∣∣∣+ ∣∣∣∣ a2a1a3∣∣∣∣+ ∣∣∣∣a1a3a2a4

∣∣∣∣+ ∣∣∣∣ a2a4a1a3a5∣∣∣∣+ · · · < +∞∑

n

χ0(SnT−1n , Sn−1T−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

A classical theorem The parabolic region The Stern–Stolz series

Thank you!

A classical theorem

The parabolic region

The Stern–Stolz series