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Picard’s theorems and Schottky’s theorem

within second order arithmetic

Yoshihiro Horihata Tohoku university

Joint work with Keita Yokoyama

27 May, 2011

Tohoku logic seminar

Today’s topics

1. Picard’s theorem via Riemann mapping theorem. 2. Picard’s theorem via Schottky’s theorem.

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Picard theorem via RMT

Definition (RCA₀)

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f : [0, 1] → R: continuous.

We say that f is effectively integrable on [0, 1] when we can find a modulus of integrability for f .

Here, modulus of integrability for f is a function h : N → N such that for all n ∈ N and for all partitions

∆1, ∆2 ^{of [0, 1],}

|∆1^|, |∆2^{| < 2−h(n) → |S ∆1( f ) − S ∆2}( f )| < 2⁻ⁿ⁺¹.

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Theorem

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The following assertions are equivalent over RCA₀. 1. WKL₀.

2. Heine/Borel theorem.

3. Every continuous function [0, 1] → R is effectively integrable.

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Theorem

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The following assertions are equivalent over RCA₀. 1. WWKL₀.

2. Weak-Heine/Borel theorem.

Here, weak-Heine/Borel theorem is following:

If [0, 1] ⊆ ^S_n∈N B(a_n; r_n) then there exists h_{(bi j, ci j)j≤li} | _{i ∈ N, li} ∈ Ni _{such that}

[0, 1] ⊆ ^[

n<i

B(a_n, r_n) ∪ ^[

j≤l_i

(b_{i j}, c_{i j})∧

i→∞lim

X

j<l_i

|_{ci j} − _{bi j| = 0.}

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Theorem

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The following assertions are equivalent over RCA₀. 1. WWKL₀.

2. Weak-Heine/Borel theorem.

3. Every bounded continuous function is effectively integrable.

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Next, we see some basic theorems of complex analysis in SOA.

Definition (holomorphic function)

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The following definition is made in RCA₀. D ⊆ C: open,

f , f ^′ : D → C: continuous.

Then a pair ( f , f ^′) is said to be holomorphic if

∀_{z ∈ D lim}

w→z

f (w) − f (z)

w − z ^{= f}

′(z).

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Theorem

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The following assertions are equivalent over RCA₀. 1. WKL₀.

2. Cauchy’s integral theorem.

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Cauchy’s integral theorem

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D ⊆ C: open,

f : D → C: holomorphic. Then, for all △abc ⊆ D, ^R

∂△abc f (z) dz exists and Z

∂△abc

f (z) dz = 0.

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RCA₀ version of Cauchy’s integral theorem

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D ⊆ C: open,

f : D → C: holomorphic and effectively integrable. Then, for all △abc ⊆ D, ^R_∂△abc f (z) dz exists and

Z

∂△abc

f (z) dz = 0.

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Theorem

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RCA₀ proves the RCA₀ version of Cauchy’s integral theorem.

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Definition

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f : D = {z | 0 ≤ R1 < |z − a| < R₂^{} → C}: holomorphic.

Then, a is said to be an isolated essential singularity if there exists {a_n^}_n∈Z such that f (z) = ^P_n∈Z ^an^{(z − a)n for}

all z ∈ D and ∀m ∈ N∃k ≥ m (a_−k ^, 0).

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Theorem

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WWKL₀ proves Riemann’s theorem on removable sin- gularities:

D := {z | 0 < |z − a| < r},

f : D → C : holomorphic.

If there exists r^′ > 0 such that r^′ < r and f is bounded on {z | 0 < |z − a| < r^′}, then there exists a holomorphic function ˜f : D ∪ {a} → C such that ˜f (z) = f (z) for all z ∈ D.

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Theorem

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WWKL₀ proves Casorati/Weierstraß theorem: D := {z | 0 < |z − a| < r},

f : D → C : holomorphic,

a is isolated essential singularity. Then f (D) is dense in C.

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Theorem

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WWKL₀ proves Casorati/Weierstraß theorem: D := {z | 0 < |z − a| < r},

f : D → C : holomorphic,

a is isolated essential singularity. Then f (D) is dense in C.

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Picard’s little theorem

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f : C → C : holomorphic.

If f is non-constant, then f attains all values of C with at most one exception.

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(1) Liouville theorem.

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(1) Liouville theorem. (2) Lifting lemma.

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(1) Liouville theorem. (2) Lifting lemma.

(3) We can take ∆(1) as a covering space of C \ {_{0, 1}.}

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(1) Liouville theorem. ^⇐ provable in WWKL₀. (2) Lifting lemma.

(3) We can take ∆(1) as a covering space of C \ {_{0, 1}.}

Lifting lemma

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(X, D, π, U_{i j}, V_i, π_{i j}) : covering space, D ⊆ C : open,

f : D₀ ^→ D : continuous.

Then, if D₀ is simply connected, then there exists a continuous function ˆf : D₀ ^→ X such that π ◦ ˆ_{f = f .}

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Lifting lemma

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(X, D, π, U_{i j}, V_i, π_{i j}) : covering space, D ⊆ C : open,

f : D₀ ^→ D : continuous.

Then, if D₀ is simply connected, then there exists a continuous function ˆf : D₀ ^→ X such that π ◦ ˆ_{f = f .}

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Theorem

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The following assertions are equivalent over RCA₀. 1. WKL₀.

2. Lifting lemma.

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(1) Liouville theorem. ⇐ provable in WWKL₀. (2) Lifting lemma. ^⇐ equivalent to WKL₀.

(3) We can take ∆(1) as a covering space of C \ {_{0, 1}.}

(3)We reason within ACA₀.

Since Riemann mapping theorem is equivalent to ACA₀ over WKL₀, we can construct a biholomorphic function:

f : Ω := {z | Im(z) > 0 ∧ −1 < Re(z) < 1 ∧ |z| > 1} → ∆(1).

−_{1 1}

Ω ^∆(1)

f

We can prove Schwarz reflection principle within WWKL₀.

We can prove Schwarz reflection principle within WWKL₀. We can construct a covering map π : ∆(1) → C \ {0, 1} by

applying Schwarz reflection principle to Ω.

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(1) Liouville theorem. ⇐ provable in WWKL₀. (2) Lifting lemma. ⇐ equivalent to WKL₀.

(3) We can take ∆(1) as a covering space of C \ {_{0, 1}.} ⇐ provable in ACA₀.

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(1) Liouville theorem. ⇐ provable in WWKL₀. (2) Lifting lemma. ⇐ equivalent to WKL₀.

(3) We can take ∆(1) as a covering space of C \ {0, 1}. ⇐ provable in ACA₀.

Therefore, we can prove Picard’s little theorem within ACA₀.

But, in (3),

−1 1

Ω ^∆(1)

f

since Ω is a simple domain, we expected that we can con- struct this biholomorphic function f within RCA₀.

Definition (semi-polygon)

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A semi-polygon is a finite sequence of functions γ = hγ₁, · · · , γ_ni _{where γi} : [(i − 1)/n, i/n] → C (1 ≤ i ≤ n) is a line or an arc of a circle, γ_{i(i/n) = γi+1}(i/n) for all 1 ≤ i ≤ n and γ_{1(0) = γn}(1).

A semi-polygon γ is said to be simple if γ(t) , γ(s) for all 0 ≤ t < s < 1.

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Theorem (weak-Riemann mapping theorem)

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The following is provable in RCA₀. γ : simple semi-polygon on C,

ϕ : linear transformation s.t. 0 ∈ ϕ(Int(γ)) ⊆ ∆(1), D := ϕ(Int(γ)).

Then, there exists a biholomorphic function f : D →

∆(1) such that f (0) = 0.

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But, to prove Picard’s theorem, we need to expand the map f into f : D → ∆(1).

This is very hard.

Theorem’ (weak-Riemann mapping theorem)

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The following is provable in RCA₀. γ : simple semi-polygon on C,

ϕ : linear transformation s.t. 0 ∈ ϕ(Int(γ)) ⊆ ∆(1), D := ϕ(Int(γ)).

Then, there exists a biholomorphic function f : D →

∆(1) such that f (0) = 0.

Moreover, we can take f such that f is uniformly continuous on D;

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Theorem’’ (weak-Riemann mapping theorem)

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The following is provable in RCA₀. γ : simple semi-polygon on C,

ϕ : linear transformation s.t. 0 ∈ ϕ(Int(γ)) ⊆ ∆(1), D := ϕ(Int(γ)).

Then, there exists a biholomorphic function f : D →

∆(1) such that f (0) = 0.

Moreover, we can take f such that f is uniformly continuous on D; f is limit-expandable into ∂D;

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Consequently,

• RMT for simple semi-polygon is provable in RCA₀.

• RMT for Jordan curve is equivalent to WKL₀.

• RMT is equivalent to ACA₀ over WKL₀.

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Picard theorem via Schottky

Lemma

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The following assertions are provable in WKL₀. Let R > 0 be a constant and f : ∆(R) → C be a holomorphic function such that f (z) , 0 on ∆(R). Then, there exist holomorphic functions h(z) and g(z) such that

f (z) = e^h(z), f (z) = (g(z))ⁿ

where n is a positive integer. Then, we denote log f (z) and ⁿ p f (z) as h(z) and g(z), respectively.

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Bloch’s theorem

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The following assertions are provable in WKL₀. Let f : ∆(1) → C be a holomorphic function.

Then, if f (0) = 0 and f ^′(0) = 1, then, ∆(B) ⊆ f (∆(1)). Here, B = 1/32.

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Schottky’s theorem

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The following assertions are provable in WKL₀. Let M > 0 and 0 < r < 1.

Then, there exists C, depending only on M and r, which satisfies the following properties:

If f : ∆(1) → C is a holomorphic function such that

| f (0)| ≤ M and f omits 0 and 1, then, | f (z)| ≤ C on

∆(r).

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Picard’s big theorem

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The following assertions are provable in WKL₀. Fix R > 0 and z₀ ^{∈ C}.

Let f : B(z₀; R) \ z₀ ^{→ C} be a holomorphic function. Then, if z₀ is an essential singularity of f , then, for each positive r < R, f assumes, on B(z₀; r) \ z₀, every value of C with at most one exception.

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We see the proof.

We reason within WKL₀.

To prove contradiction, assume that f omits two points. By translation, we may assume that z₀ = 0, f is holomor- phic on ∆(2π) \ 0 and omits 0, 1. Since 0 is an essentially singularity of f , lim_z→0 ^| f (z)| does not converge (also, is not ∞).

Hence, there exists M > 0 such that for each n, there ex- ists x ∈ Q + iQ such that |x| < 2⁻ⁿ and | f (x)| < M.

We may assume M > | f (1/2)|.

Then, we want to find the sequence hz_n ^| n ∈ Ni satisfies the following:

• ^limn→∞ ^zn = 0;

• ^{1 > |z}1^{| > |z}2^| > · · · > |z_n^| > · · · ;

• ^{| f (z}n)| ≤ M for each n ∈ N.

To find the sequence, we use primitive recursion.

By primitive recursion, we can define h : N → Q + iQ as

h(0) = ¹ 2^;

h(n + 1) = λxψ(x, h(n))

where ψ(x, h(n)) ≡ x ∈ Q+iQ∧| f (x)| < M ∧|x| < min{h(n), 2^−n} is Σ⁰₁ ^formula.

Write ψ(x, h(n)) ≡ ∃kθ(k, x, h(n)), where θ is Σ⁰₀ ^formula. Then, λxψ(x, h(n)) := (µl θ[(l)0^{, (l)}1^{, h(n)])}1^.

Then, let hz_n | n ∈ Ni be a sequence such that z_{n = h(n) for} each n ∈ N.

Then, hz_n ^| n ∈ Ni satisfies the following:

• ^limn→∞ ^zn = 0;

• ^{1 > |z}1^{| > |z}2^| > · · · > |z_n^| > · · · ;

• ^{| f (z}n)| ≤ M for each n ∈ N.

Fix j ∈ N. Define a function G : ∆(1) → C as G(ζ) := f (zj exp(2πiζ)).

Then, G is holomorphic on ∆(1), omits 0, 1, and satisfies

|_{G(0)| ≤ M.}

Hence, by Schottky’s theorem, there exists K, which de- pends only on M, such that

|G(ζ)| = | f (zj exp(2πiζ))| ≤ K (ζ ∈ ∆(1/2)).

This especially implies that

| _{f (zj} exp(2πit))| ≤ K (−1/2 ≤ t ≤ 1/2). Hence, | f (z)| ≤ K on |z| = |zj^|.

This holds for each j ∈ N, since K depends only on M.

|_{z j}|

|_zj+1|

D_j

By the maximum modulus principle, f is bounded by K on each D_j := {z ∈ C | |zj^{| ≤ |z| ≤ |Z} _j+1^|}.

Then, f is bounded by K on ∆(|z1^|) \ {0}, since for each point z ∈ ∆(|z1^|) belongs in D_j for some j ∈ N.

Therefore, by Riemann’s theorem on removable sin- gularities, 0 is a removable singularity, and hence con- tradiction.

This completes the proof.

Picard’s little theorem

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The following assertions are provable in WKL₀. f : C → C : holomorphic.

If f is non-constant, then f attains all values of C with at most one exception.

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Proof of Picard’s little theorem

Assume that f : C → C is non-constant function.

Put g(z) := f (1/z). If z = 0 is an essential singularity of g, then it follows from Big Picard’s theorem.

If z = 0 is a pole or removable singularity of g, then g(z) − P

n≤N ^anz−n is entire and bounded.

Here, ^P_n≤N a_nz⁻ⁿ is the principal part of g.

Then, by Liouville’s theorem, g(z) − ^P_n≤N a_nz⁻ⁿ _{= C for} some constant C ∈ C, and hence f is a polynomial. Hence, by the fundamental theorem of algebra, f attains all val- ues of C with at most one exception.

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Future studies

• ^RCA0 proves the extension version of Weak RMT?? Study the next theorem within second order arithmetic.

Theorem (K. T. Hahn, 1983)

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Let D ⊆ C be a domain.

Then, the following assertions are equivalent: 1. D has at least two boundary points.

2. D has the Picard property. 3. D has the Schottky property. 4. D has the Landau property.

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