Constructing Entire Functions (a summary) Kirill Lazebnik SUNY - - PowerPoint PPT Presentation

Constructing Entire Functions (a summary)

Kirill Lazebnik

SUNY Stony Brook Kirill.Lazebnik@stonybrook.edu

November 26, 2015

Kirill Lazebnik

Constructing Entire Functions By Quasiconformal Folding (a summary)

Kirill Lazebnik

SUNY Stony Brook Kirill.Lazebnik@stonybrook.edu

November 26, 2015

Kirill Lazebnik

p(z) = z3

2 − 3z 2

Kirill Lazebnik

p(z) = z3

2 − 3z 2

p′(z) = 3

2(z − 1)(z + 1)

Kirill Lazebnik

p(z) = z3

2 − 3z 2 (two critical values ±1)

p′(z) = 3

2(z − 1)(z + 1)

Kirill Lazebnik

p(z) = z3

2 − 3z 2 (two critical values ±1)

p′(z) = 3

2(z − 1)(z + 1)

Kirill Lazebnik

p(z) = z4

4 − z3 3 − z2 2 + z

Kirill Lazebnik

p(z) = z4

4 − z3 3 − z2 2 + z

p′(z) = (z − 1)2(z + 1)

Kirill Lazebnik

p(z) = z4

4 − z3 3 − z2 2 + z (two critical values 5/12, −11/12)

p′(z) = (z − 1)2(z + 1)

Kirill Lazebnik

p(z) = z4

4 − z3 3 − z2 2 + z (two critical values 5/12, −11/12)

p′(z) = (z − 1)2(z + 1)

Kirill Lazebnik

Shabat polynomial -

Kirill Lazebnik

Shabat polynomial - only has two critical values ±1

Kirill Lazebnik

Shabat polynomial - only has two critical values ±1 Proposition: For any Shabat polynomial p(z), it is true that p−1[−1, 1] is a tree.

Kirill Lazebnik

Shabat polynomial - only has two critical values ±1 Proposition: For any Shabat polynomial p(z), it is true that p−1[−1, 1] is a tree, with deg(p) edges.

Kirill Lazebnik

Shabat polynomial - only has two critical values ±1 Proposition: For any Shabat polynomial p(z), it is true that p−1[−1, 1] is a tree, with deg(p) edges. Theorem (Grothendieck): ALL combinatorial trees occur as p−1[−1, 1] for some Shabat polynomial p(z).

Kirill Lazebnik

Shabat polynomial - only has two critical values ±1 Proposition: For any Shabat polynomial p(z), it is true that p−1[−1, 1] is a tree, with deg(p) edges. Theorem (Grothendieck): ALL combinatorial trees occur as p−1[−1, 1] for some Shabat polynomial p(z). Theorem (Bishop): Any continua can be ǫ-approximated in the Hausdorff metric by some p−1[−1, 1].

Kirill Lazebnik

trees ⇐ ⇒ Shabat polynomials

Kirill Lazebnik

trees ⇐ ⇒ Shabat polynomials infinite trees ⇐ ⇒ Transcendental Functions

Kirill Lazebnik

trees ⇐ ⇒ Shabat polynomials infinite trees ⇐ ⇒ Subclass of Transcendental Functions

Kirill Lazebnik

trees ⇐ ⇒ Shabat polynomials infinite trees ⇐ ⇒ Subclass of Transcendental Functions S2,0 - transcendental functions with two critical values ±1 and no asymptotic values

Kirill Lazebnik

cosh(z) := ez+e−z

2

Kirill Lazebnik

cosh(z) := ez+e−z

2

cosh′(z) = ez−e−z

2

Kirill Lazebnik

cosh(z) := ez+e−z

2

cosh′(z) = ez−e−z

2

= 0 = ⇒ z = πin : n ∈ Z

Kirill Lazebnik

cosh(z) := ez+e−z

2

cosh′(z) = ez−e−z

2

= 0 = ⇒ z = πin : n ∈ Z (critical points)

Kirill Lazebnik

cosh(z) := ez+e−z

2

cosh′(z) = ez−e−z

2

= 0 = ⇒ z = πin : n ∈ Z (critical points) critical values: ±1

Kirill Lazebnik

T - unbounded, locally finite tree

Kirill Lazebnik

T - unbounded, locally finite tree, with a bipartite labeling of vertices.

Kirill Lazebnik

T - unbounded, locally finite tree, with a bipartite labeling of vertices. Ωj - components of C − T

Kirill Lazebnik

T - unbounded, locally finite tree, with a bipartite labeling of vertices. Ωj - components of C − T τ : ∪Ωj → C - the map conformal on each Ωj to Hr

Kirill Lazebnik

T - unbounded, locally finite tree, with a bipartite labeling of vertices. Ωj - components of C − T τ : ∪Ωj → C - the map conformal on each Ωj to Hr V - the vertices of T.

Kirill Lazebnik

T - unbounded, locally finite tree, with a bipartite labeling of vertices. Ωj - components of C − T. τ : ∪Ωj → C - the map conformal on each Ωj to Hr. V - the vertices of T. Vj - the image of V under τ restricted to Ωj.

Kirill Lazebnik

T - unbounded, locally finite tree, with a bipartite labeling of vertices. Ωj - components of C − T. τ : ∪Ωj → C - the map conformal on each Ωj to Hr. V - the vertices of T. Vj - the image of V under τ restricted to Ωj. For r > 0, define T(r) = ∪e∈T{z : dist(z, e) < rdiam(e)}

Kirill Lazebnik

T - unbounded, locally finite tree, with a bipartite labeling of vertices. Ωj - components of C − T. τ : ∪Ωj → C - the map conformal on each Ωj to Hr. V - the vertices of T. Vj - the image of V under τ restricted to Ωj. For r > 0, define T(r) = ∪e∈T{z : dist(z, e) < rdiam(e)} The τ-size of edge e is the minimum length of the two images τ(e)

Kirill Lazebnik

T - unbounded, locally finite tree, with a bipartite labeling of vertices. Ωj - components of C − T. τ : ∪Ωj → C - the map conformal on each Ωj to Hr. V - the vertices of T. Vj - the image of V under τ restricted to Ωj. For r > 0, define T(r) = ∪e∈T{z : dist(z, e) < rdiam(e)} The τ-size of edge e is the minimum length of the two images τ(e) T has uniformly bounded geometry if:

Kirill Lazebnik

T - unbounded, locally finite tree, with a bipartite labeling of vertices. Ωj - components of C − T. τ : ∪Ωj → C - the map conformal on each Ωj to Hr. V - the vertices of T. Vj - the image of V under τ restricted to Ωj. For r > 0, define T(r) = ∪e∈T{z : dist(z, e) < rdiam(e)} The τ-size of edge e is the minimum length of the two images τ(e) T has uniformly bounded geometry if: (1) The edges of T are C 2 with uniform bounds.

Kirill Lazebnik

T - unbounded, locally finite tree, with a bipartite labeling of vertices. Ωj - components of C − T. τ : ∪Ωj → C - the map conformal on each Ωj to Hr. V - the vertices of T. Vj - the image of V under τ restricted to Ωj. For r > 0, define T(r) = ∪e∈T{z : dist(z, e) < rdiam(e)} The τ-size of edge e is the minimum length of the two images τ(e) T has uniformly bounded geometry if: (1) The edges of T are C 2 with uniform bounds. (2) The angles between adjacent edges are bounded uniformly from zero

Kirill Lazebnik

T - unbounded, locally finite tree, with a bipartite labeling of vertices. Ωj - components of C − T. τ : ∪Ωj → C - the map conformal on each Ωj to Hr. V - the vertices of T. Vj - the image of V under τ restricted to Ωj. For r > 0, define T(r) = ∪e∈T{z : dist(z, e) < rdiam(e)} The τ-size of edge e is the minimum length of the two images τ(e) T has uniformly bounded geometry if: (1) The edges of T are C 2 with uniform bounds. (2) The angles between adjacent edges are bounded uniformly from zero (3) Adjacent edges have uniformly comparable lengths

Kirill Lazebnik

T - unbounded, locally finite tree, with a bipartite labeling of vertices. Ωj - components of C − T. τ : ∪Ωj → C - the map conformal on each Ωj to Hr. V - the vertices of T. Vj - the image of V under τ restricted to Ωj. For r > 0, define T(r) = ∪e∈T{z : dist(z, e) < rdiam(e)} The τ-size of edge e is the minimum length of the two images τ(e) T has uniformly bounded geometry if: (1) The edges of T are C 2 with uniform bounds. (2) The angles between adjacent edges are bounded uniformly from zero (3) Adjacent edges have uniformly comparable lengths (4) For non-adjacent edges e, f , we have diam(e)/dist(e, f ) uniformly bounded.

Kirill Lazebnik

T - unbounded, locally finite tree, with a bipartite labeling of vertices. Ωj - components of C − T. τ : ∪Ωj → C - the map conformal on each Ωj to Hr . V - the vertices of T. Vj - the image of V under τ restricted to Ωj . For r > 0, define T(r) = ∪e∈T {z : dist(z, e) < rdiam(e)} The τ-size of edge e is the minimum length of the two images τ(e) T has uniformly bounded geometry if: (1) The edges of T are C2 with uniform bounds. (2) The angles between adjacent edges are bounded uniformly from zero (3) Adjacent edges have uniformly comparable lengths (4) For non-adjacent edges e, f , we have diam(e)/dist(e, f ) uniformly bounded.

Theorem:

Kirill Lazebnik

T - unbounded, locally finite tree, with a bipartite labeling of vertices. Ωj - components of C − T. τ : ∪Ωj → C - the map conformal on each Ωj to Hr . V - the vertices of T. Vj - the image of V under τ restricted to Ωj . For r > 0, define T(r) = ∪e∈T {z : dist(z, e) < rdiam(e)} The τ-size of edge e is the minimum length of the two images τ(e) T has uniformly bounded geometry if: (1) The edges of T are C2 with uniform bounds. (2) The angles between adjacent edges are bounded uniformly from zero (3) Adjacent edges have uniformly comparable lengths (4) For non-adjacent edges e, f , we have diam(e)/dist(e, f ) uniformly bounded.

Theorem: Suppose T has bounded geometry

Kirill Lazebnik

T - unbounded, locally finite tree, with a bipartite labeling of vertices. Ωj - components of C − T. τ : ∪Ωj → C - the map conformal on each Ωj to Hr . V - the vertices of T. Vj - the image of V under τ restricted to Ωj . For r > 0, define T(r) = ∪e∈T {z : dist(z, e) < rdiam(e)} The τ-size of edge e is the minimum length of the two images τ(e) T has uniformly bounded geometry if: (1) The edges of T are C2 with uniform bounds. (2) The angles between adjacent edges are bounded uniformly from zero (3) Adjacent edges have uniformly comparable lengths (4) For non-adjacent edges e, f , we have diam(e)/dist(e, f ) uniformly bounded.

Theorem: Suppose T has bounded geometry and every edge has τ-size ≥ π.

Kirill Lazebnik

T - unbounded, locally finite tree, with a bipartite labeling of vertices. Ωj - components of C − T. τ : ∪Ωj → C - the map conformal on each Ωj to Hr . V - the vertices of T. Vj - the image of V under τ restricted to Ωj . For r > 0, define T(r) = ∪e∈T {z : dist(z, e) < rdiam(e)} The τ-size of edge e is the minimum length of the two images τ(e) T has uniformly bounded geometry if: (1) The edges of T are C2 with uniform bounds. (2) The angles between adjacent edges are bounded uniformly from zero (3) Adjacent edges have uniformly comparable lengths (4) For non-adjacent edges e, f , we have diam(e)/dist(e, f ) uniformly bounded.

Theorem: Suppose T has bounded geometry and every edge has τ-size ≥ π. Then there is an r0 > 0, an entire f , and a K-quasiconformal φ

Kirill Lazebnik

T - unbounded, locally finite tree, with a bipartite labeling of vertices. Ωj - components of C − T. τ : ∪Ωj → C - the map conformal on each Ωj to Hr . V - the vertices of T. Vj - the image of V under τ restricted to Ωj . For r > 0, define T(r) = ∪e∈T {z : dist(z, e) < rdiam(e)} The τ-size of edge e is the minimum length of the two images τ(e) T has uniformly bounded geometry if: (1) The edges of T are C2 with uniform bounds. (2) The angles between adjacent edges are bounded uniformly from zero (3) Adjacent edges have uniformly comparable lengths (4) For non-adjacent edges e, f , we have diam(e)/dist(e, f ) uniformly bounded.

Theorem: Suppose T has bounded geometry and every edge has τ-size ≥ π. Then there is an r0 > 0, an entire f , and a K-quasiconformal φ so that f ◦ φ = cosh ◦τ off T(r0).

Kirill Lazebnik

T - unbounded, locally finite tree, with a bipartite labeling of vertices. Ωj - components of C − T. τ : ∪Ωj → C - the map conformal on each Ωj to Hr . V - the vertices of T. Vj - the image of V under τ restricted to Ωj . For r > 0, define T(r) = ∪e∈T {z : dist(z, e) < rdiam(e)} The τ-size of edge e is the minimum length of the two images τ(e) T has uniformly bounded geometry if: (1) The edges of T are C2 with uniform bounds. (2) The angles between adjacent edges are bounded uniformly from zero (3) Adjacent edges have uniformly comparable lengths (4) For non-adjacent edges e, f , we have diam(e)/dist(e, f ) uniformly bounded.

Theorem: Suppose T has bounded geometry and every edge has τ-size ≥ π. Then there is an r0 > 0, an entire f , and a K-quasiconformal φ so that f ◦ φ = cosh ◦τ off T(r0). K depends

• nly on the bounded geometry constants of T.

Kirill Lazebnik

T - unbounded, locally finite tree, with a bipartite labeling of vertices. Ωj - components of C − T. τ : ∪Ωj → C - the map conformal on each Ωj to Hr . V - the vertices of T. Vj - the image of V under τ restricted to Ωj . For r > 0, define T(r) = ∪e∈T {z : dist(z, e) < rdiam(e)} The τ-size of edge e is the minimum length of the two images τ(e) T has uniformly bounded geometry if: (1) The edges of T are C2 with uniform bounds. (2) The angles between adjacent edges are bounded uniformly from zero (3) Adjacent edges have uniformly comparable lengths (4) For non-adjacent edges e, f , we have diam(e)/dist(e, f ) uniformly bounded.

Theorem: Suppose T has bounded geometry and every edge has τ-size ≥ π. Then there is an r0 > 0, an entire f , and a K-quasiconformal φ so that f ◦ φ = cosh ◦τ off T(r0). K depends

• nly on the bounded geometry constants of T. The only critical

values of f are ±1 and f has no asymptotic values.

Kirill Lazebnik

f : C → C entire function

Kirill Lazebnik

f : C → C entire function f ◦n is normal in an open set U

Kirill Lazebnik

f : C → C entire function f ◦n is normal in an open set U if every sequence of f ◦k contains a further subsequence converging locally uniformly to a holomorphic function g : U → C

Kirill Lazebnik

f : C → C entire function f ◦n is normal in an open set U if every sequence of f ◦k contains a further subsequence converging locally uniformly to a holomorphic function g : U → C The Fatou set of f is the set of points z ∈ C for which f is normal in some neighborhood of z.

Kirill Lazebnik

f : C → C entire function f ◦n is normal in an open set U if every sequence of f ◦k contains a further subsequence converging locally uniformly to a holomorphic function g : U → C The Fatou set of f is the set of points z ∈ C for which f is normal in some neighborhood of z. The components of the Fatou set are called Fatou components.

Kirill Lazebnik

f : C → C entire function f ◦n is normal in an open set U if every sequence of f ◦k contains a further subsequence converging locally uniformly to a holomorphic function g : U → C The Fatou set of f is the set of points z ∈ C for which f is normal in some neighborhood of z. The components of the Fatou set are called Fatou components. A Fatou component U is called wandering if f n(U) ∩ f m(U) = ∅ for all n, m ∈ N, n = m

Kirill Lazebnik

Theorem: (Sullivan) Rational maps don’t have wandering domains.

Kirill Lazebnik

Theorem: (Sullivan) Rational maps don’t have wandering domains. For f : C → C, the singular set S(f ) consists of the critical values and asymptotic values of f .

Kirill Lazebnik

Theorem: (Sullivan) Rational maps don’t have wandering domains. For f : C → C, the singular set S(f ) consists of the critical values and asymptotic values of f . The Speiser class S consists of those transcendental functions for which S(f ) is finite.

Kirill Lazebnik

Theorem: (Sullivan) Rational maps don’t have wandering domains. For f : C → C, the singular set S(f ) consists of the critical values and asymptotic values of f . The Speiser class S consists of those transcendental functions for which S(f ) is finite. Theorem: (Golberg and Keen, Eremenko and Lyubich) Functions in the Speiser Class don’t have wandering domains.

Kirill Lazebnik

Theorem: (Sullivan) Rational maps don’t have wandering domains. For f : C → C, the singular set S(f ) consists of the critical values and asymptotic values of f . The Speiser class S consists of those transcendental functions for which S(f ) is finite. Theorem: (Golberg and Keen, Eremenko and Lyubich) Functions in the Speiser Class don’t have wandering domains. The Eremenko-Lyubich class B consists of those transcendental functions with bounded singular set.

Kirill Lazebnik

S+ = {x + iy : x > 0, |y| < π/2}

Kirill Lazebnik

S+ = {x + iy : x > 0, |y| < π/2} is mapped conformally to Hr by λ · sinh.

Kirill Lazebnik

S+ = {x + iy : x > 0, |y| < π/2} is mapped conformally to Hr by λ · sinh. Then holomorphically to C − [−1, 1] by cosh.

Kirill Lazebnik

S+ = {x + iy : x > 0, |y| < π/2} is mapped conformally to Hr by λ · sinh. Then holomorphically to C − [−1, 1] by cosh. an = cosh−1 π

λ

λ

π cosh(nπ)

• Kirill Lazebnik

S+ = {x + iy : x > 0, |y| < π/2} is mapped conformally to Hr by λ · sinh. Then holomorphically to C − [−1, 1] by cosh. an = cosh−1 π

λ

λ

π cosh(nπ)

• zn = an + iπ

Kirill Lazebnik

S+ = {x + iy : x > 0, |y| < π/2} is mapped conformally to Hr by λ · sinh. Then holomorphically to C − [−1, 1] by cosh. an = cosh−1 π

λ

λ

π cosh(nπ)

• zn = an + iπ, Dn = {z ∈ C : |z − zn| < 1}

Kirill Lazebnik

S+ = {x + iy : x > 0, |y| < π/2} is mapped conformally to Hr by λ · sinh. Then holomorphically to C − [−1, 1] by cosh. an = cosh−1 π

λ

λ

π cosh(nπ)

• zn = an + iπ, Dn = {z ∈ C : |z − zn| < 1} is mapped

holomorphically to |z| < 1 by z → (z − zn)dn.

Kirill Lazebnik

S+ = {x + iy : x > 0, |y| < π/2} is mapped conformally to Hr by λ · sinh. Then holomorphically to C − [−1, 1] by cosh. an = cosh−1 π

λ

λ

π cosh(nπ)

• zn = an + iπ, Dn = {z ∈ C : |z − zn| < 1} is mapped

holomorphically to |z| < 1 by z → (z − zn)dn. Then by a quasiconformal map ρ of the disc so that:

Kirill Lazebnik

S+ = {x + iy : x > 0, |y| < π/2} is mapped conformally to Hr by λ · sinh. Then holomorphically to C − [−1, 1] by cosh. an = cosh−1 π

λ

λ

π cosh(nπ)

• zn = an + iπ, Dn = {z ∈ C : |z − zn| < 1} is mapped

holomorphically to |z| < 1 by z → (z − zn)dn. Then by a quasiconformal map ρn of the disc so that: (1) ρn(z) = z for z ∈ ∂D

Kirill Lazebnik

S+ = {x + iy : x > 0, |y| < π/2} is mapped conformally to Hr by λ · sinh. Then holomorphically to C − [−1, 1] by cosh. an = cosh−1 π

λ

λ

π cosh(nπ)

• zn = an + iπ, Dn = {z ∈ C : |z − zn| < 1} is mapped

holomorphically to |z| < 1 by z → (z − zn)dn. Then by a quasiconformal map ρn of the disc so that: (1) ρn(z) = z for z ∈ ∂D (2) ρn(0) = wn where wn is a point near 1/2.

Kirill Lazebnik

S+ = {x + iy : x > 0, |y| < π/2} is mapped conformally to Hr by λ · sinh. Then holomorphically to C − [−1, 1] by cosh. an = cosh−1 π

λ

λ

π cosh(nπ)

• zn = an + iπ, Dn = {z ∈ C : |z − zn| < 1} is mapped

holomorphically to |z| < 1 by z → (z − zn)dn. Then by a quasiconformal map ρn of the disc so that: (1) ρn(z) = z for z ∈ ∂D (2) ρn(0) = wn where wn is a point near 1/2. (3) ρn is conformal on 3

4D

Kirill Lazebnik

S+ = {x + iy : x > 0, |y| < π/2} is mapped conformally to Hr by λ · sinh. Then holomorphically to C − [−1, 1] by cosh. an = cosh−1 π

λ

λ

π cosh(nπ)

• zn = an + iπ, Dn = {z ∈ C : |z − zn| < 1} is mapped

holomorphically to |z| < 1 by z → (z − zn)dn. Then by a quasiconformal map ρn of the disc so that: (1) ρn(z) = z for z ∈ ∂D (2) ρn(0) = wn where wn is a point near 1/2. (3) ρn is conformal on 3

4D

(4) ρn is quasiconformal on D.

Kirill Lazebnik

S+ = {x + iy : x > 0, |y| < π/2} is mapped conformally to Hr by λ · sinh. Then holomorphically to C − [−1, 1] by cosh. an = cosh−1

π λ

• λ

π cosh(nπ)

• zn = an + iπ, Dn = {z ∈ C : |z − zn| < 1} is mapped holomorphically to |z| < 1 by z → (z − zn)dn . Then

by a quasiconformal map ρn of the disc so that: (1) ρn(z) = z for z ∈ ∂D (2) ρn(0) = wn where wn is a point near 1/2. (3) ρn is conformal on 3

4 D

(4) ρn is quasiconformal on D.

Theorem:

Kirill Lazebnik

S+ = {x + iy : x > 0, |y| < π/2} is mapped conformally to Hr by λ · sinh. Then holomorphically to C − [−1, 1] by cosh. an = cosh−1

π λ

• λ

π cosh(nπ)

• zn = an + iπ, Dn = {z ∈ C : |z − zn| < 1} is mapped holomorphically to |z| < 1 by z → (z − zn)dn . Then

by a quasiconformal map ρn of the disc so that: (1) ρn(z) = z for z ∈ ∂D (2) ρn(0) = wn where wn is a point near 1/2. (3) ρn is conformal on 3

4 D

(4) ρn is quasiconformal on D.

Theorem: For every choice of parameters λ, (dn), (wn) with λ ∈ πN, dn ∈ 2N

Kirill Lazebnik

S+ = {x + iy : x > 0, |y| < π/2} is mapped conformally to Hr by λ · sinh. Then holomorphically to C − [−1, 1] by cosh. an = cosh−1

π λ

• λ

π cosh(nπ)

• zn = an + iπ, Dn = {z ∈ C : |z − zn| < 1} is mapped holomorphically to |z| < 1 by z → (z − zn)dn . Then

by a quasiconformal map ρn of the disc so that: (1) ρn(z) = z for z ∈ ∂D (2) ρn(0) = wn where wn is a point near 1/2. (3) ρn is conformal on 3

4 D

(4) ρn is quasiconformal on D.

Theorem: For every choice of parameters λ, (dn), (wn) with λ ∈ πN, dn ∈ 2N, there exists a transcendental f and a quasiconformal φ : C → C so that:

Kirill Lazebnik

S+ = {x + iy : x > 0, |y| < π/2} is mapped conformally to Hr by λ · sinh. Then holomorphically to C − [−1, 1] by cosh. an = cosh−1

π λ

• λ

π cosh(nπ)

• zn = an + iπ, Dn = {z ∈ C : |z − zn| < 1} is mapped holomorphically to |z| < 1 by z → (z − zn)dn . Then

by a quasiconformal map ρn of the disc so that: (1) ρn(z) = z for z ∈ ∂D (2) ρn(0) = wn where wn is a point near 1/2. (3) ρn is conformal on 3

4 D

(4) ρn is quasiconformal on D.

Theorem: For every choice of parameters λ, (dn), (wn) with λ ∈ πN, dn ∈ 2N, there exists a transcendental f and a quasiconformal φ : C → C so that: (1) f (z) = f (z), f (−z) = f (z)

Kirill Lazebnik

S+ = {x + iy : x > 0, |y| < π/2} is mapped conformally to Hr by λ · sinh. Then holomorphically to C − [−1, 1] by cosh. an = cosh−1

π λ

• λ

π cosh(nπ)

• zn = an + iπ, Dn = {z ∈ C : |z − zn| < 1} is mapped holomorphically to |z| < 1 by z → (z − zn)dn . Then

by a quasiconformal map ρn of the disc so that: (1) ρn(z) = z for z ∈ ∂D (2) ρn(0) = wn where wn is a point near 1/2. (3) ρn is conformal on 3

4 D

(4) ρn is quasiconformal on D.

Theorem: For every choice of parameters λ, (dn), (wn) with λ ∈ πN, dn ∈ 2N, there exists a transcendental f and a quasiconformal φ : C → C so that: (1) f (z) = f (z), f (−z) = f (z) (2) f (z) =

• cosh(λ sinh(φ(z)))

if φ(z) ∈ S+ ρn((φ(z) − zn)dn) if φ(z) ∈ Dn

Kirill Lazebnik

S+ = {x + iy : x > 0, |y| < π/2} is mapped conformally to Hr by λ · sinh. Then holomorphically to C − [−1, 1] by cosh. an = cosh−1

π λ

• λ

π cosh(nπ)

• zn = an + iπ, Dn = {z ∈ C : |z − zn| < 1} is mapped holomorphically to |z| < 1 by z → (z − zn)dn . Then

by a quasiconformal map ρn of the disc so that: (1) ρn(z) = z for z ∈ ∂D (2) ρn(0) = wn where wn is a point near 1/2. (3) ρn is conformal on 3

4 D

(4) ρn is quasiconformal on D.

Theorem: For every choice of parameters λ, (dn), (wn) with λ ∈ πN, dn ∈ 2N, there exists a transcendental f and a quasiconformal φ : C → C so that: (1) f (z) = f (z), f (−z) = f (z) (2) f (z) =

• cosh(λ sinh(φ(z)))

if φ(z) ∈ S+ ρn((φ(z) − zn)dn) if φ(z) ∈ Dn (3) f has no asymptotic values and its set of critical values is ±1 and {wn : n ≥ 1}

Kirill Lazebnik

S+ = {x + iy : x > 0, |y| < π/2} is mapped conformally to Hr by λ · sinh. Then holomorphically to C − [−1, 1] by cosh. an = cosh−1

π λ

• λ

π cosh(nπ)

• zn = an + iπ, Dn = {z ∈ C : |z − zn| < 1} is mapped holomorphically to |z| < 1 by z → (z − zn)dn . Then

by a quasiconformal map ρn of the disc so that: (1) ρn(z) = z for z ∈ ∂D (2) ρn(0) = wn where wn is a point near 1/2. (3) ρn is conformal on 3

4 D

(4) ρn is quasiconformal on D.

Theorem: For every choice of parameters λ, (dn), (wn) with λ ∈ πN, dn ∈ 2N, there exists a transcendental f and a quasiconformal φ : C → C so that: (1) f (z) = f (z), f (−z) = f (z) (2) f (z) =

• cosh(λ sinh(φ(z)))

if φ(z) ∈ S+ ρn((φ(z) − zn)dn) if φ(z) ∈ Dn (3) f has no asymptotic values and its set of critical values is ±1 and {wn : n ≥ 1} (4) φ(0) = 0, φ(R) = R and φ is conformal in S+.

Kirill Lazebnik

References

Chris Bishop (2014) Constructing Entire Functions By Quasiconformal Folding Acta Mathematica Nuria Fagella, Sebastien Godillion, and Xavier Jarque (2014) Wandering domains for composition of entire functions Journal of Mathematical Analysis and Applications

Kirill Lazebnik