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