Can you compute a conformal map? Ilia Binder University of Toronto - - PowerPoint PPT Presentation

Can you compute a conformal map?

Ilia Binder

University of Toronto

Based on joint work with M. Braverman (Princeton), C. Rojas (Universidad Andres Bello), and M. Yampolsky (University of Toronto)

Modern Aspects of Complex Analysis and Its Applications August 20, 2019

How to compute a conformal map?

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Computability: a crash course Inside the domain: computability and complexity Boundary behaviour: Caratheodory extension Boundary behaviour: harmonic measure

Computability of natural numbers and functions on natural numbers

A function f : N → N is called computable if there exists an algorithm (a Turing machine) which, upon input n, outputs f(n).

1

Computability of natural numbers and functions on natural numbers

A function f : N → N is called computable if there exists an algorithm (a Turing machine) which, upon input n, outputs f(n). A set E ⊆ N is called computable if its characteristic function is computable.

1

Computability of natural numbers and functions on natural numbers

A function f : N → N is called computable if there exists an algorithm (a Turing machine) which, upon input n, outputs f(n). A set E ⊆ N is called computable if its characteristic function is computable. An explicit example of non-computable set: Halting set H: i ∈ H iff the i-th Turing machine halts.

1

Computability of natural numbers and functions on natural numbers

A function f : N → N is called computable if there exists an algorithm (a Turing machine) which, upon input n, outputs f(n). A set E ⊆ N is called computable if its characteristic function is computable. An explicit example of non-computable set: Halting set H: i ∈ H iff the i-th Turing machine halts. A set E ⊆ N is called lower-computable if there exists an algorithm that enumerates E, i.e. on an input n it halts if n ∈ E, and never halts

• therwise.

1

Computability of natural numbers and functions on natural numbers

A function f : N → N is called computable if there exists an algorithm (a Turing machine) which, upon input n, outputs f(n). A set E ⊆ N is called computable if its characteristic function is computable. An explicit example of non-computable set: Halting set H: i ∈ H iff the i-th Turing machine halts. A set E ⊆ N is called lower-computable if there exists an algorithm that enumerates E, i.e. on an input n it halts if n ∈ E, and never halts

• therwise. The algorithm can verify the inclusion n ∈ E, but not the

inclusion n ∈ Ec.

1

Computability of natural numbers and functions on natural numbers

A function f : N → N is called computable if there exists an algorithm (a Turing machine) which, upon input n, outputs f(n). A set E ⊆ N is called computable if its characteristic function is computable. An explicit example of non-computable set: Halting set H: i ∈ H iff the i-th Turing machine halts. A set E ⊆ N is called lower-computable if there exists an algorithm that enumerates E, i.e. on an input n it halts if n ∈ E, and never halts

• therwise. The algorithm can verify the inclusion n ∈ E, but not the

inclusion n ∈ Ec. Halting set is a classical example of a lower-computable non computable set.

1

Computability of natural numbers and functions on natural numbers

A function f : N → N is called computable if there exists an algorithm (a Turing machine) which, upon input n, outputs f(n). A set E ⊆ N is called computable if its characteristic function is computable. An explicit example of non-computable set: Halting set H: i ∈ H iff the i-th Turing machine halts. A set E ⊆ N is called lower-computable if there exists an algorithm that enumerates E, i.e. on an input n it halts if n ∈ E, and never halts

• therwise. The algorithm can verify the inclusion n ∈ E, but not the

inclusion n ∈ Ec. Halting set is a classical example of a lower-computable non computable set. A complement of lower-computable set is called upper-computable.

1

Computability of natural numbers and functions on natural numbers

A function f : N → N is called computable if there exists an algorithm (a Turing machine) which, upon input n, outputs f(n). A set E ⊆ N is called computable if its characteristic function is computable. An explicit example of non-computable set: Halting set H: i ∈ H iff the i-th Turing machine halts. A set E ⊆ N is called lower-computable if there exists an algorithm that enumerates E, i.e. on an input n it halts if n ∈ E, and never halts

• therwise. The algorithm can verify the inclusion n ∈ E, but not the

inclusion n ∈ Ec. Halting set is a classical example of a lower-computable non computable set. A complement of lower-computable set is called upper-computable. A set is computable iff it is simultaneously upper- and lower-computable.

1

Computability of reals and functions

x ∈ R is called

• computable if there is a computable function f : N → Q such that

|f(n) − x| < 2−n;

• lower-computable if there is a computable function f : N → Q such

that f(n) ↑ x;

• upper-computable if there is a computable function f : N → Q such

that f(n) ↓ x.

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Computability of reals and functions

x ∈ R is called

• computable if there is a computable function f : N → Q such that

|f(n) − x| < 2−n;

• lower-computable if there is a computable function f : N → Q such

that f(n) ↑ x;

• upper-computable if there is a computable function f : N → Q such

that f(n) ↓ x. A function φ : N → Q2 is an oracle for x ∈ C if |φ(n) − x| < 2−n.

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Computability of reals and functions

x ∈ R is called

• computable if there is a computable function f : N → Q such that

|f(n) − x| < 2−n;

• lower-computable if there is a computable function f : N → Q such

that f(n) ↑ x;

• upper-computable if there is a computable function f : N → Q such

that f(n) ↓ x. A function φ : N → Q2 is an oracle for x ∈ C if |φ(n) − x| < 2−n. A function f : S → C (S ⊂ C) is called computable if there exists an algorithm with an oracle for x ∈ S and an input n ∈ N which outputs a rational point sn such that |sn − f(x)| < 2−n. An algorithm may query an oracle by reading the values of the function φ for an arbitrary m ∈ N.

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Computability of planar sets.

Let Bn be an enumeration of all planar rational balls. Lower-computable open set. An open set U is called lower-computable if there is a computable function f : N → N such that U =

n∈N Bf(n).

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Computability of planar sets.

Let Bn be an enumeration of all planar rational balls. Lower-computable open set. An open set U is called lower-computable if there is a computable function f : N → N such that U =

n∈N Bf(n).

Lower-computable closed set. Closed set K is called lower computable if the set S ⊂ N with i ∈ S iff Bi ∩ K = ∅ is lower-computable.

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Computability of planar sets.

Let Bn be an enumeration of all planar rational balls. Lower-computable open set. An open set U is called lower-computable if there is a computable function f : N → N such that U =

n∈N Bf(n).

Lower-computable closed set. Closed set K is called lower computable if the set S ⊂ N with i ∈ S iff Bi ∩ K = ∅ is lower-computable. A compact is computable iff it and its complement are lower-computable.

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Computability of planar sets.

Let Bn be an enumeration of all planar rational balls. Lower-computable open set. An open set U is called lower-computable if there is a computable function f : N → N such that U =

n∈N Bf(n).

Lower-computable closed set. Closed set K is called lower computable if the set S ⊂ N with i ∈ S iff Bi ∩ K = ∅ is lower-computable. A compact is computable iff it and its complement are lower-computable. Equivalently, K ⊂ Rd is computable if there exists an algorithm A with a single input n ∈ N which outputs a finite set Cn of points with rational coordinates (or a rational polygon) such that Hdist(Cn, K) < 2−n.

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Computability of planar sets.

Let Bn be an enumeration of all planar rational balls. Lower-computable open set. An open set U is called lower-computable if there is a computable function f : N → N such that U =

n∈N Bf(n).

Lower-computable closed set. Closed set K is called lower computable if the set S ⊂ N with i ∈ S iff Bi ∩ K = ∅ is lower-computable. A compact is computable iff it and its complement are lower-computable. Equivalently, K ⊂ Rd is computable if there exists an algorithm A with a single input n ∈ N which outputs a finite set Cn of points with rational coordinates (or a rational polygon) such that Hdist(Cn, K) < 2−n. Hdist(X, Y ) = max

• supx∈X dist(x, Y ), supy∈Y dist(y, X)
• .

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Computing the Riemann map

Let Ω C be a simply connected planar domain with w0 ∈ Ω. g = gΩ,w0 is the unique conformal map of Ω onto the unit disk D with g(w0) = 0, g′(w0) > 0. f := g−1.

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Computing the Riemann map

Let Ω C be a simply connected planar domain with w0 ∈ Ω. g = gΩ,w0 is the unique conformal map of Ω onto the unit disk D with g(w0) = 0, g′(w0) > 0. f := g−1. Constructive Riemann Mapping Theorem.(Hertling, 1997) The following are equivalent: (i) Ω is a lower-computable open set, ∂Ω is a lower-computable closed set, and w0 ∈ Ω is a computable point; (ii) The maps g and f are both computable conformal bijections.

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Computing the Riemann map: idea of the proof.

(i) = ⇒ (ii): if Ω is a lower-computable open set, ∂Ω is a lower-computable closed set, and w0 ∈ Ω is a computable point then both f and g are computable.

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Computing the Riemann map: idea of the proof.

(i) = ⇒ (ii): if Ω is a lower-computable open set, ∂Ω is a lower-computable closed set, and w0 ∈ Ω is a computable point then both f and g are computable. The lower-computability of Ω implies that one can compute a sequence

• f polygonal domains with rational vertices Ωn such that Ω = ∪Ωn The

maps fn : D → Ωn are explicitly computable (by Schwarz-Christoffel, for example) and converge to f. To check that fn(z) approximates f(z) well enough, we just need to approximate the boundary from below by centers

• f rational balls intersecting it. Then fn is close to f because the

corresponding domains are close in Caratheodory sense.

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Computing the Riemann map: idea of the proof.

(i) = ⇒ (ii): if Ω is a lower-computable open set, ∂Ω is a lower-computable closed set, and w0 ∈ Ω is a computable point then both f and g are computable. The lower-computability of Ω implies that one can compute a sequence

• f polygonal domains with rational vertices Ωn such that Ω = ∪Ωn The

maps fn : D → Ωn are explicitly computable (by Schwarz-Christoffel, for example) and converge to f. To check that fn(z) approximates f(z) well enough, we just need to approximate the boundary from below by centers

• f rational balls intersecting it. Then fn is close to f because the

corresponding domains are close in Caratheodory sense. (ii) = ⇒ (i): just follows from Distortion Theorems and Caratheodory Convergence Theorem.

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Hierarchy of Complexity Classes

Question: How hard is it to compute a conformal map g in a given point w ∈ Ω?

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Hierarchy of Complexity Classes

Question: How hard is it to compute a conformal map g in a given point w ∈ Ω? P – computable in time polynomial in the length of the input. NP – solution can be checked in polynomial time. #P – can be reduced to counting the number of satisfying assignments for a given propositional formula (#SAT). PSPACE – solvable in space polynomial in the input size. EXP – solvable in time 2nc for some c (n – the length of input).

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Hierarchy of Complexity Classes

Question: How hard is it to compute a conformal map g in a given point w ∈ Ω? P – computable in time polynomial in the length of the input. NP – solution can be checked in polynomial time. #P – can be reduced to counting the number of satisfying assignments for a given propositional formula (#SAT). PSPACE – solvable in space polynomial in the input size. EXP – solvable in time 2nc for some c (n – the length of input). KNOWN: P = EXP. CONJECTURED: P NP #P PSPACE EXP.

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An upper bound on computational complexity

Theorem (B-Braverman-Yampolsky). There is an algorithm A that computes the uniformizing map in the following sense: Let Ω be a bounded simply-connected domain, and w0 ∈ Ω. Assume that the boundary of a simply connected domain Ω, ∂Ω, w0 ∈ Ω, and w ∈ Ω are provided to A by an oracle. Then A computes g(w) with precision n with complexity PSPACE(n).

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An upper bound on computational complexity

Theorem (B-Braverman-Yampolsky). There is an algorithm A that computes the uniformizing map in the following sense: Let Ω be a bounded simply-connected domain, and w0 ∈ Ω. Assume that the boundary of a simply connected domain Ω, ∂Ω, w0 ∈ Ω, and w ∈ Ω are provided to A by an oracle. Then A computes g(w) with precision n with complexity PSPACE(n). The algorithm uses solution of Dirichlet problem with random walk and de-randomization.

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An upper bound on computational complexity

Theorem (B-Braverman-Yampolsky). There is an algorithm A that computes the uniformizing map in the following sense: Let Ω be a bounded simply-connected domain, and w0 ∈ Ω. Assume that the boundary of a simply connected domain Ω, ∂Ω, w0 ∈ Ω, and w ∈ Ω are provided to A by an oracle. Then A computes g(w) with precision n with complexity PSPACE(n). The algorithm uses solution of Dirichlet problem with random walk and de-randomization. Later improved by Rettinger to #P.

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A lower bound on computational complexity

Theorem (B-Braverman-Yampolsky). Suppose there is an algorithm A that given a simply-connected domain Ω with a linear-time computable boundary, a point w0 ∈ Ω with dist(w0, ∂Ω) > 1

2 and a number n,

computes 20n digits of the conformal radius f′(0)), then we can use one call to A to solve any instance of a #SAT(n) with a linear time overhead. In other words, #P is poly-time (in fact, linear time) reducible to computing the conformal radius of a set. Any algorithm computing values of the uniformization map will also compute the conformal radius with the same precision, by Distortion Theorem.

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Proof of the lower bound

For a propositional formula Φ with n variables, let L ⊂ {0, 1, . . . , 2n − 1} be the set of numbers corresponding to its satisfying instances. Let k be the number of elements of L.

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Proof of the lower bound

For a propositional formula Φ with n variables, let L ⊂ {0, 1, . . . , 2n − 1} be the set of numbers corresponding to its satisfying instances. Let k be the number of elements of L. Let ΩL be defined as D \ ∪l∈L{|z − exp(2πil2−n)| ≤ 2−10n}, the unit disk with k very small and spaced out half balls removed.

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Proof of the lower bound

For a propositional formula Φ with n variables, let L ⊂ {0, 1, . . . , 2n − 1} be the set of numbers corresponding to its satisfying instances. Let k be the number of elements of L. Let ΩL be defined as D \ ∪l∈L{|z − exp(2πil2−n)| ≤ 2−10n}, the unit disk with k very small and spaced out half balls removed. The key estimate: if f : (D, 0) → (ΩL, 0) is conformal, f′(0) > 0 and n is large enough, then

• f′(0) − 1 + k2−20n−1

< 1 1002−20n.

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Proof of the lower bound

For a propositional formula Φ with n variables, let L ⊂ {0, 1, . . . , 2n − 1} be the set of numbers corresponding to its satisfying instances. Let k be the number of elements of L. Let ΩL be defined as D \ ∪l∈L{|z − exp(2πil2−n)| ≤ 2−10n}, the unit disk with k very small and spaced out half balls removed. The key estimate: if f : (D, 0) → (ΩL, 0) is conformal, f′(0) > 0 and n is large enough, then

• f′(0) − 1 + k2−20n−1

< 1 1002−20n. The boundary of ΩL is computable in linear time, given the access to Φ. The estimate implies that using the algorithm A we can evaluate |L| = k, and solve the #SAT problem on Φ.

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Carath´ eodory extension

Carath´ eodory extension of f to ∂Ω is given by Carth´ eodory Theorem (Theorem 1.3.1 in ): Let Ω ⊂ C be a simply-connected domain. A conformal map f : D → Ω extends to a continuous map D → Ω iff ∂Ω is locally connected.

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Carath´ eodory extension

Carath´ eodory extension of f to ∂Ω is given by Carth´ eodory Theorem (Theorem 1.3.1 in ): Let Ω ⊂ C be a simply-connected domain. A conformal map f : D → Ω extends to a continuous map D → Ω iff ∂Ω is locally connected. A set K ⊂ C is called locally connected if there exists modulus of local connectivity m(δ): a non-decreasing function decaying to 0 as δ → 0 and such that for any x, y ∈ K with |x − y| < δ one can find a connected C ⊂ K containing x and y with diam C < m(δ).

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Carath´ eodory extension

Carath´ eodory extension of f to ∂Ω is given by Carth´ eodory Theorem (Theorem 1.3.1 in ): Let Ω ⊂ C be a simply-connected domain. A conformal map f : D → Ω extends to a continuous map D → Ω iff ∂Ω is locally connected. A set K ⊂ C is called locally connected if there exists modulus of local connectivity m(δ): a non-decreasing function decaying to 0 as δ → 0 and such that for any x, y ∈ K with |x − y| < δ one can find a connected C ⊂ K containing x and y with diam C < m(δ). f extends to a homeomorphism D → Ω iff ∂Ω is a Jordan curve.

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Carath´ eodory extension.

What information about Ω does one need to compute f up to the boundary (i.e. the Caratheodory Extension)?

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Carath´ eodory extension.

What information about Ω does one need to compute f up to the boundary (i.e. the Caratheodory Extension)? Natural to assume that m(δ) for ∂Ω has to be computable.

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Carath´ eodory extension.

What information about Ω does one need to compute f up to the boundary (i.e. the Caratheodory Extension)? Natural to assume that m(δ) for ∂Ω has to be computable. Wrong!

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Carath´ eodory extension.

What information about Ω does one need to compute f up to the boundary (i.e. the Caratheodory Extension)? Natural to assume that m(δ) for ∂Ω has to be computable. Wrong! Carath´ eodory modulus. A non-decreasing function η(δ) is called the Carath´ eodory modulus of Ω if η(δ) → 0 as δ → 0 and if for every crosscut γ with diam(γ) < δ we have diam Nγ < η(δ). Here Nγ is the component of Ω \ γ not containing w0.

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Carath´ eodory extension.

What information about Ω does one need to compute f up to the boundary (i.e. the Caratheodory Extension)? Natural to assume that m(δ) for ∂Ω has to be computable. Wrong! Carath´ eodory modulus. A non-decreasing function η(δ) is called the Carath´ eodory modulus of Ω if η(δ) → 0 as δ → 0 and if for every crosscut γ with diam(γ) < δ we have diam Nγ < η(δ). Here Nγ is the component of Ω \ γ not containing w0. η(δ) ≤ m(δ), but η(δ) exists iff m(δ) exists.

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Carath´ eodory extension.

What information about Ω does one need to compute f up to the boundary (i.e. the Caratheodory Extension)? Natural to assume that m(δ) for ∂Ω has to be computable. Wrong! Carath´ eodory modulus. A non-decreasing function η(δ) is called the Carath´ eodory modulus of Ω if η(δ) → 0 as δ → 0 and if for every crosscut γ with diam(γ) < δ we have diam Nγ < η(δ). Here Nγ is the component of Ω \ γ not containing w0. η(δ) ≤ m(δ), but η(δ) exists iff m(δ) exists. Closer related to the Modulus of local connectivity m′(δ) of C \ Ω: m′(δ) ≤ 2η(δ) + δ.

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Carath´ eodory extension.

What information about Ω does one need to compute f up to the boundary (i.e. the Caratheodory Extension)? Natural to assume that m(δ) for ∂Ω has to be computable. Wrong! Carath´ eodory modulus. A non-decreasing function η(δ) is called the Carath´ eodory modulus of Ω if η(δ) → 0 as δ → 0 and if for every crosscut γ with diam(γ) < δ we have diam Nγ < η(δ). Here Nγ is the component of Ω \ γ not containing w0. η(δ) ≤ m(δ), but η(δ) exists iff m(δ) exists. Closer related to the Modulus of local connectivity m′(δ) of C \ Ω: m′(δ) ≤ 2η(δ) + δ. Theorem(B-Rojas-Yampolsky) The Carath´ eodory extension of f : D → Ω is computable iff f is computable and there exists a computable Carath´ eodory modulus of Ω. Furthermore, there exists a domain Ω with computable Carath´ eodory modulus but no computable modulus of local connectivity.

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General simply-connected domains: Carath´ eodory metric.

Carth´ eodory metric on (Ω, w): distC(z1, z2) = inf diam(γ), where γ is a closed curve or crosscut in Ω separating {z1, z2} from w0. (Defined as continuous extension when one of the points is equal to w0.)

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General simply-connected domains: Carath´ eodory metric.

Carth´ eodory metric on (Ω, w): distC(z1, z2) = inf diam(γ), where γ is a closed curve or crosscut in Ω separating {z1, z2} from w0. (Defined as continuous extension when one of the points is equal to w0.) The closure of Ω in Carath´ eodory metric is called the Carath´ eodory compactification, ˆ Ω. It is obtained from Ω by adding the prime ends.

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General simply-connected domains: Carath´ eodory metric.

Carth´ eodory metric on (Ω, w): distC(z1, z2) = inf diam(γ), where γ is a closed curve or crosscut in Ω separating {z1, z2} from w0. (Defined as continuous extension when one of the points is equal to w0.) The closure of Ω in Carath´ eodory metric is called the Carath´ eodory compactification, ˆ Ω. It is obtained from Ω by adding the prime ends. Carath´ eodory Theorem: f is extendable to a homeomorphism ˆ f : D → ˆ Ω.

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General simply-connected domains: Carath´ eodory metric.

Carth´ eodory metric on (Ω, w): distC(z1, z2) = inf diam(γ), where γ is a closed curve or crosscut in Ω separating {z1, z2} from w0. (Defined as continuous extension when one of the points is equal to w0.) The closure of Ω in Carath´ eodory metric is called the Carath´ eodory compactification, ˆ Ω. It is obtained from Ω by adding the prime ends. Carath´ eodory Theorem: f is extendable to a homeomorphism ˆ f : D → ˆ Ω. Computable Carath´ eodory Theorem (B-Rojas-Yampolsky): In the presence of oracles for w0 and for ∂Ω, both ˆ f and ˆ g = ˆ f−1 are computable.

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Warshawski’s Theorem

Oscillation of f near boundary. ω(r) := sup

|z0|=1,|z1|<1, |z2|<1,|z1−z0|<r,|z2−z0|<r

|f(z1) − f(z2)|.

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Warshawski’s Theorem

Oscillation of f near boundary. ω(r) := sup

|z0|=1,|z1|<1, |z2|<1,|z1−z0|<r,|z2−z0|<r

|f(z1) − f(z2)|. Allows the estimate |f(z) − f((1 − r)z)| ≤ ω(r) for |z| = 1.

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Warshawski’s Theorem

Oscillation of f near boundary. ω(r) := sup

|z0|=1,|z1|<1, |z2|<1,|z1−z0|<r,|z2−z0|<r

|f(z1) − f(z2)|. Allows the estimate |f(z) − f((1 − r)z)| ≤ ω(r) for |z| = 1. Warshawski’s Theorem (1950) ω(r) ≤ η 2πA log 1/r 1/2 , for all r ∈ (0, 1). A is the area of Ω, and η(δ) is Caratheodory modulus.

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Warshawski’s Theorem

Oscillation of f near boundary. ω(r) := sup

|z0|=1,|z1|<1, |z2|<1,|z1−z0|<r,|z2−z0|<r

|f(z1) − f(z2)|. Allows the estimate |f(z) − f((1 − r)z)| ≤ ω(r) for |z| = 1. Warshawski’s Theorem (1950) ω(r) ≤ η 2πA log 1/r 1/2 , for all r ∈ (0, 1). A is the area of Ω, and η(δ) is Caratheodory modulus. Basically, a restatement of classical Wolff’s Lemma(1934), Equation (5.11) in Let |z0| = 1, and let kr be the arc of the circle |z − z0| = r which is contained in D. Then for every r ∈ (0, 1) there exists ρ∗ ∈ (r, √r) such that the image of kρ∗ is a crosscut of W of length at most

• 2πA

log 1/r

1/2 .

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Warshawski’s Theorem: a proof

Wolff’s Lemma is easily provable by a length-area argument.

14

Warshawski’s Theorem: a proof

Wolff’s Lemma is easily provable by a length-area argument. Derivation of Warshawski’s Theorem from Wolff’s lemma For a z0 with |z0| = 1, let Tr := f({|z − z0| < r, |z| < 1}) ⊂ Ω.

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Warshawski’s Theorem: a proof

Wolff’s Lemma is easily provable by a length-area argument. Derivation of Warshawski’s Theorem from Wolff’s lemma For a z0 with |z0| = 1, let Tr := f({|z − z0| < r, |z| < 1}) ⊂ Ω.ω(r) = supz0 diam(Tr).

14

Warshawski’s Theorem: a proof

Wolff’s Lemma is easily provable by a length-area argument. Derivation of Warshawski’s Theorem from Wolff’s lemma For a z0 with |z0| = 1, let Tr := f({|z − z0| < r, |z| < 1}) ⊂ Ω.ω(r) = supz0 diam(Tr). Select ρ∗ ∈ (r, √r) from Wolff’s Lemma. Then Tr ⊂ Tρ∗.

14

Warshawski’s Theorem: a proof

Wolff’s Lemma is easily provable by a length-area argument. Derivation of Warshawski’s Theorem from Wolff’s lemma For a z0 with |z0| = 1, let Tr := f({|z − z0| < r, |z| < 1}) ⊂ Ω.ω(r) = supz0 diam(Tr). Select ρ∗ ∈ (r, √r) from Wolff’s Lemma. Then Tr ⊂ Tρ∗. The image γ := f(kρ∗) is a crosscut of W with diam γ ≤ ℓρ∗. Hence, by definition

• f a Carath´

eodory modulus, diam Tr ≤ diam Tρ∗ ≤ η 2πA log 1/r 1/2 .

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Other direction: Lavrentieff-type estimate

A refinement of Lavrentieff estimate(1936), Corollary IV.9.3 in : Let M = dist(∂Ω, w0), γ be a crosscut with dist(∂Ω, w0) ≥ M/2, ǫ2 < M/4. Then diam(γ) < ǫ2 = ⇒ diam(f−1(Nγ)) ≤ 30ǫ √ M .

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Other direction: Lavrentieff-type estimate

A refinement of Lavrentieff estimate(1936), Corollary IV.9.3 in : Let M = dist(∂Ω, w0), γ be a crosscut with dist(∂Ω, w0) ≥ M/2, ǫ2 < M/4. Then diam(γ) < ǫ2 = ⇒ diam(f−1(Nγ)) ≤ 30ǫ √ M . Essentially, ˆ f−1 is 1/2-H¨

• lder as a map from ˆ

Ω to D.

15

Other direction: Lavrentieff-type estimate

A refinement of Lavrentieff estimate(1936), Corollary IV.9.3 in : Let M = dist(∂Ω, w0), γ be a crosscut with dist(∂Ω, w0) ≥ M/2, ǫ2 < M/4. Then diam(γ) < ǫ2 = ⇒ diam(f−1(Nγ)) ≤ 30ǫ √ M . Essentially, ˆ f−1 is 1/2-H¨

• lder as a map from ˆ

Ω to D. Corollary η(ǫ2) ≤ 2ω 30ǫ √ M

• .

15

Other direction: Lavrentieff-type estimate

A refinement of Lavrentieff estimate(1936), Corollary IV.9.3 in : Let M = dist(∂Ω, w0), γ be a crosscut with dist(∂Ω, w0) ≥ M/2, ǫ2 < M/4. Then diam(γ) < ǫ2 = ⇒ diam(f−1(Nγ)) ≤ 30ǫ √ M . Essentially, ˆ f−1 is 1/2-H¨

• lder as a map from ˆ

Ω to D. Corollary η(ǫ2) ≤ 2ω 30ǫ √ M

• .

Proof. diam(Nγ) ≤ 2ω(diam(f−1(Nγ))) ≤ 2ω 30ǫ √ M

• .

15

A domain with computable Carath´ eodory extension and no computable modulus of local connectivity: construction

Let again B ⊂ N be a lower-computable, non-computable set. Set xi = 1 − 1/2i.

16

A domain with computable Carath´ eodory extension and no computable modulus of local connectivity: construction

Let again B ⊂ N be a lower-computable, non-computable set. Set xi = 1 − 1/2i. The domain Ω is constructed by modifying the square (0, 1) × (0, 1) as follows.

16

A domain with computable Carath´ eodory extension and no computable modulus of local connectivity: construction

Let again B ⊂ N be a lower-computable, non-computable set. Set xi = 1 − 1/2i. The domain Ω is constructed by modifying the square (0, 1) × (0, 1) as follows.

1 1 x

j

xi x

j

xi

If i / ∈ B, then we add a straight line (i-line) to I going from (xi, 1) to (xi, xi).

16

A domain with computable Carath´ eodory extension and no computable modulus of local connectivity: construction

Let again B ⊂ N be a lower-computable, non-computable set. Set xi = 1 − 1/2i. The domain Ω is constructed by modifying the square (0, 1) × (0, 1) as follows.

1 1 x

j

xi x

j

xi

If i / ∈ B, then we add a straight line (i-line) to I going from (xi, 1) to (xi, xi). If i ∈ B and it is enumerated in stage s, we remove i-fjord, i.e. the rectangle [(xi − si, (xi + si] × [xi, 1] where si = min{2−s, 1/(3i2)}.

16

The example: ∂Ω and Carath´ eodory modulus are computable.

1 1 x

j

xi x

j

xi

Computing a 2−s Hausdorff approximation of ∂Ω. Run an algorithm enumerating B for s + 1

• steps. For all those i’s that have

been enumerated so far, draw the corresponding i-fjords. For all the

• ther i’s, draw a i-line.

17

The example: ∂Ω and Carath´ eodory modulus are computable.

1 1 x

j

xi x

j

xi

Computing a 2−s Hausdorff approximation of ∂Ω. Run an algorithm enumerating B for s + 1

• steps. For all those i’s that have

been enumerated so far, draw the corresponding i-fjords. For all the

• ther i’s, draw a i-line.

Carath´ eodory modulus: 2√r.

17

The example: Modulus of local connectivity m(r) is not computable

1 1 x

j

xi x

j

xi

Compute B using m(r).

18

The example: Modulus of local connectivity m(r) is not computable

1 1 x

j

xi x

j

xi

Compute B using m(r). First, for i ∈ N, compute ri ∈ Q such that m(2 · 2−ri) < xi 2 .

18

The example: Modulus of local connectivity m(r) is not computable

1 1 x

j

xi x

j

xi

Compute B using m(r). First, for i ∈ N, compute ri ∈ Q such that m(2 · 2−ri) < xi 2 . If i ∈ B then i is enumerated in fewer than ri steps. Our algorithm to compute B will emulate the algorithm for enumerating B for ri steps.

18

Computability of harmonic measure: simply connected case

A measure µ on a metric space X is called computable if for any computable function φ, the integral

• X φ dµ is computable.

19

Computability of harmonic measure: simply connected case

A measure µ on a metric space X is called computable if for any computable function φ, the integral

• X φ dµ is computable.

Theorem (B-Rojas-Yampolsky). Let Ω be a simply connected domain, w0 ∈ Ω be a computable point. Then the following are equivalent: (i) Ω is a lower-computable open set, ∂Ω is a lower-computable closed set; (ii) The Riemann maps g and f are both computable conformal bijections; (iii) The harmonic measure ωΩ(w0, ·) is computable.

19

Caratheodory convergence and weak convergence of harmonic measure are the same

Proof based on Theorem (B-Rojas-Yampolsky). For simply connected domains Ωn ∋ wn, n ∈ N, and Ω ∋ w, the following are equivalent: (i) (Ωn, wn) → (Ω, w) in Carath´ eodory sense, i.e. fn → f uniformly on compact subsets of D.

20

Caratheodory convergence and weak convergence of harmonic measure are the same

Proof based on Theorem (B-Rojas-Yampolsky). For simply connected domains Ωn ∋ wn, n ∈ N, and Ω ∋ w, the following are equivalent: (i) (Ωn, wn) → (Ω, w) in Carath´ eodory sense, i.e. fn → f uniformly on compact subsets of D. (ii) (Ωn, wn) and (Ω, w) have arbitrarily good common interior approximations, i.e. for every ε > 0, there exists N ∈ N and a closed connected set Kε ⊂ Ω ∩

n≥N Ωn containing w such that

dist(x, ∂Ω) < ε and dist(x, ∂Ωn) < ε holds for all x ∈ ∂Kε and all n ≥ N.

20

Caratheodory convergence and weak convergence of harmonic measure are the same

Proof based on Theorem (B-Rojas-Yampolsky). For simply connected domains Ωn ∋ wn, n ∈ N, and Ω ∋ w, the following are equivalent: (i) (Ωn, wn) → (Ω, w) in Carath´ eodory sense, i.e. fn → f uniformly on compact subsets of D. (ii) (Ωn, wn) and (Ω, w) have arbitrarily good common interior approximations, i.e. for every ε > 0, there exists N ∈ N and a closed connected set Kε ⊂ Ω ∩

n≥N Ωn containing w such that

dist(x, ∂Ω) < ε and dist(x, ∂Ωn) < ε holds for all x ∈ ∂Kε and all n ≥ N. (iii) ωΩn(wn, ·) → ωΩ(w, ·) weakly.

20

Computability of harmonic measure: general case

Theorem (B-Braverman-Rojas-Yampolsky). If a closed set K ⊂ C is computable, uniformly perfect, and has a connected complement, then in the presence of oracle for w0 / ∈ K, the harmonic measure of Ω = ˆ C \ K at w0 is computable.

21

Computability of harmonic measure: general case

Theorem (B-Braverman-Rojas-Yampolsky). If a closed set K ⊂ C is computable, uniformly perfect, and has a connected complement, then in the presence of oracle for w0 / ∈ K, the harmonic measure of Ω = ˆ C \ K at w0 is computable. A compact set K ⊂ C which contains at least two points is uniformly perfect if there exists some C > 0 such that for any x ∈ K and r > 0, we have (B(x, Cr) \ B(x, r)) ∩ K = ∅ = ⇒ K ⊂ B(x, r). In particular, every connected set is uniformly perfect.

21

Approximating harmonic measure: capacity density condition.

Theorem (Pommerenke, 1979), Exercise IX.3 of : For a domain with uniformly perfect boundary there exists a constant ν = ν(C) < 1 such that for any y ∈ Ω P[|By

T − y| ≥ 2 dist(y, ∂Ω)] < ν.

Here By

T is the first hitting of the boundary by Brownian motion started

at y.

22

Approximating harmonic measure: capacity density condition.

Theorem (Pommerenke, 1979), Exercise IX.3 of : For a domain with uniformly perfect boundary there exists a constant ν = ν(C) < 1 such that for any y ∈ Ω P[|By

T − y| ≥ 2 dist(y, ∂Ω)] < ν.

Here By

T is the first hitting of the boundary by Brownian motion started

at y. By the strong Markov property of the Brownian motion, for any n P

• |By

T − y| ≥ 2n dist(y, ∂Ω)

• < νn.

22

Approximating harmonic measure: capacity density condition.

Theorem (Pommerenke, 1979), Exercise IX.3 of : For a domain with uniformly perfect boundary there exists a constant ν = ν(C) < 1 such that for any y ∈ Ω P[|By

T − y| ≥ 2 dist(y, ∂Ω)] < ν.

Here By

T is the first hitting of the boundary by Brownian motion started

at y. By the strong Markov property of the Brownian motion, for any n P

• |By

T − y| ≥ 2n dist(y, ∂Ω)

• < νn.

Take any computable φ. We need to compute E(φ(BT )).

22

Approximating harmonic measure: capacity density condition.

Theorem (Pommerenke, 1979), Exercise IX.3 of : For a domain with uniformly perfect boundary there exists a constant ν = ν(C) < 1 such that for any y ∈ Ω P[|By

T − y| ≥ 2 dist(y, ∂Ω)] < ν.

Here By

T is the first hitting of the boundary by Brownian motion started

at y. By the strong Markov property of the Brownian motion, for any n P

• |By

T − y| ≥ 2n dist(y, ∂Ω)

• < νn.

Take any computable φ. We need to compute E(φ(BT )). Compute the interior polygonal δ-approximation Ω′ to Ω for small enough δ. Then it is easy to see that E(φ(BT ) − φ(BT ′)) is small, since with high probability BT is close to BT ′.

22

Harmonic measure: even more general condition

Modulus of uniform perfectness: ε(δ) := {ε > 0 | dist(z, ∂Ωn) < ε = ⇒ ωΩn(D(z, δ)) > 1 − δ} .

23

Harmonic measure: even more general condition

Modulus of uniform perfectness: ε(δ) := {ε > 0 | dist(z, ∂Ωn) < ε = ⇒ ωΩn(D(z, δ)) > 1 − δ} . Boundness modulus for w ∈ Ω: R(ε) :=

• R > 0 | ωΩ∩B(0,2R)(w, B(0, R)) > 1 − ε
• 23

Harmonic measure: even more general condition

Modulus of uniform perfectness: ε(δ) := {ε > 0 | dist(z, ∂Ωn) < ε = ⇒ ωΩn(D(z, δ)) > 1 − δ} . Boundness modulus for w ∈ Ω: R(ε) :=

• R > 0 | ωΩ∩B(0,2R)(w, B(0, R)) > 1 − ε
• Theorem (B-Rojas-Yampolsky). If Ω ⊂ Rd is a domain with

computable boundary, then the harmonic measure ωΩ(w0, ·) is computable for any computable point w0 ∈ Ω if and only if it has a computable modulus of uniform regularity and a computable boundness modulus.

23

Harmonic measure: even more general condition

Modulus of uniform perfectness: ε(δ) := {ε > 0 | dist(z, ∂Ωn) < ε = ⇒ ωΩn(D(z, δ)) > 1 − δ} . Boundness modulus for w ∈ Ω: R(ε) :=

• R > 0 | ωΩ∩B(0,2R)(w, B(0, R)) > 1 − ε
• Theorem (B-Rojas-Yampolsky). If Ω ⊂ Rd is a domain with

computable boundary, then the harmonic measure ωΩ(w0, ·) is computable for any computable point w0 ∈ Ω if and only if it has a computable modulus of uniform regularity and a computable boundness modulus. For Ω ⊂ ˆ C, the boundness modulus is unnecessary.

23

A domain with computable boundary and noncomputable harmonic measure.

Let B ⊂ N be a lower-computable, non-computable set. We modify the unit circle by inserting the following ”gates” at exp 2πi (2−n):

e

2-n

• 2n

i π 2 ( ) 2

• e

2-n

• 2n

i π 2 ( ) 2 +

e

2-n

• 2n

i π 2 ( ) 2

• e

2-n

• 2n

i π 2 ( ) 2 +

Ln

8 j

Ln

24

A domain with computable boundary and noncomputable harmonic measure.

Let B ⊂ N be a lower-computable, non-computable set. We modify the unit circle by inserting the following ”gates” at exp 2πi (2−n):

e

2-n

• 2n

i π 2 ( ) 2

• e

2-n

• 2n

i π 2 ( ) 2 +

e

2-n

• 2n

i π 2 ( ) 2

• e

2-n

• 2n

i π 2 ( ) 2 +

Ln

8 j

Ln

Specifically, if n ∈ B is enumerated at stage j we take the interval [exp 2πi

• 2−n − 2−2n

, exp 2πi

• 2−n + 2−2n

] and insert j equally spaced small arcs such that the harmonic measure of the ”outer part of the gate” is at least 1/2 × 2−2n, producing a j-gate.

24

A domain with computable boundary and noncomputable harmonic measure.

Let B ⊂ N be a lower-computable, non-computable set. We modify the unit circle by inserting the following ”gates” at exp 2πi (2−n):

e

2-n

• 2n

i π 2 ( ) 2

• e

2-n

• 2n

i π 2 ( ) 2 +

e

2-n

• 2n

i π 2 ( ) 2

• e

2-n

• 2n

i π 2 ( ) 2 +

Ln

8 j

Ln

Specifically, if n ∈ B is enumerated at stage j we take the interval [exp 2πi

• 2−n − 2−2n

, exp 2πi

• 2−n + 2−2n

] and insert j equally spaced small arcs such that the harmonic measure of the ”outer part of the gate” is at least 1/2 × 2−2n, producing a j-gate. Otherwise, if n / ∈ B, we almost cover the gate with one interval so that the harmonic measure on the the ”outer part of the gate” is at most 2−100n, making an ∞-gate.

24

A domain with computable boundary and noncomputable harmonic measure.

e

2-n

• 2n

i π 2 ( ) 2

• e

2-n

• 2n

i π 2 ( ) 2 +

e

2-n

• 2n

i π 2 ( ) 2

• e

2-n

• 2n

i π 2 ( ) 2 +

Ln

8 j

Ln

The resulting domain Ω is regular.

25

A domain with computable boundary and noncomputable harmonic measure.

e

2-n

• 2n

i π 2 ( ) 2

• e

2-n

• 2n

i π 2 ( ) 2 +

e

2-n

• 2n

i π 2 ( ) 2

• e

2-n

• 2n

i π 2 ( ) 2 +

Ln

8 j

Ln

The resulting domain Ω is regular. To compute its boundary with precision 1/j, run an algorithm enumerating B for j steps. Insert j-gate for all n which are not yet enumerated.

25

A domain with computable boundary and noncomputable harmonic measure.

e

2-n

• 2n

i π 2 ( ) 2

• e

2-n

• 2n

i π 2 ( ) 2 +

e

2-n

• 2n

i π 2 ( ) 2

• e

2-n

• 2n

i π 2 ( ) 2 +

Ln

8 j

Ln

The resulting domain Ω is regular. To compute its boundary with precision 1/j, run an algorithm enumerating B for j steps. Insert j-gate for all n which are not yet enumerated. But if the harmonic measure of Ω would be computable, we would just have to compute it with precision 2−10n to decide if n ∈ B. This contradicts non-computability of B!

25

Thank you for your attention.

26