Bishop's qc-folding and wandering domains in Eremenko ... - ICMS

Bishop’s qc-folding and wandering
domains in Eremenko-Lyubich class
—
X. Jarque
Universitat de Barcelona
—
ICMS, Edinburgh, Scotland, UK
23 May, 2013
Wandering domains and Bishop’s folding

Background
Definition: Let f be a rational or entire transcendental map. Let U
be a Fatou domain of f . If f l (U), l ≥ 0 is never eventually
periodic then we say that U is wandering. In this case we have
f n (U) ∩ f m (U) = ∅ ∀n < m ∈ Z.
Theorem (Sullivan 1985): Let R : Ĉ → Ĉ be a rational map and
let U be a Fatou domain of R. Then f l (U) is eventually periodic
for some l ≥ 0. In other words, U cannot be a wandering domain.
Wandering domains and Bishop’s folding

Background
Definition: Let f be a rational or entire transcendental map. Let U
be a Fatou domain of f . If f l (U), l ≥ 0 is never eventually
periodic then we say that U is wandering. In this case we have
f n (U) ∩ f m (U) = ∅ ∀n < m ∈ Z.
Theorem (Sullivan 1985): Let R : Ĉ → Ĉ be a rational map and
let U be a Fatou domain of R. Then f l (U) is eventually periodic
for some l ≥ 0. In other words, U cannot be a wandering domain.
Remark: In what follows f will be an entire transcendental map.
Wandering domains and Bishop’s folding

Multiply-connected wandering domain (Baker 1976)
∏
∞
Let g(z) = Cz
(1 2 + z )
where
a n
n=1
1 < a 1 < a 2 < . . .,
a j+1 < g(a j ) < 2a n+1 ,
g(A j ) ⊂ A j+1 where
A j = {z ∈ C | a 2 j < |z| < √ a j+1 }.
Remark: If A j ⊂ U j then g(U j ) = U j+1 , hence g k (U j ) → ∞ as
k → ∞ for all j.
Theorem (Baker 1975, 1985): If U is a multiply-connected
component of the Fatou set of f then U wanders.
Wandering domains and Bishop’s folding

Wandering domain example II (Herman-Sullivan’s 80’s)
C
⏐ ⏐↓
e −z
z−1+e −z +2πi
−−−−−−−−−→
C
⏐ ⏐↓e
−z
(1)
C \ {0}
ewe −w
−−−−→ C \ {0}
Lemma: If f and g are entire, f and g commutes, and f = g + c
for some constant c, then J(f ) = J(g).
Application: f (z) = z − 1 + e −z and g(z) = f (z) + 2πi. So, the
Fatou set of g has a wandering domain.
Wandering domains and Bishop’s folding

Wandering domain example III
Let g(z) = z + λ 0 sin(z) with λ 0 ≈ 6.36227.
The (infinitely many) critical points: x = cos −1 (−1/λ 0 ).
λ 0 is such that g(x) = x + 2πi for all c.p. x.
Only two critical orbits, both belonging to the Fatou set.
The lines {x = kπ} ⊂ J (g) for all k ∈ Z.
x 6.36227 sinx
30
20
10
30 20 10 10 20 30
x
10
20
30
Wandering domains and Bishop’s folding

Critically finite entire transcendental functions
We denote by S(f ) the set of (finite) singularities of f −1 (critical
values, asymptotic values and limits of those values).
Definition: We say that f is critically finite if S(f ) is finite.
E λ (z) = λexp(z)
Sn λ (z) = λ sin(z)
Theorem (Baker 1984): If f (z) =
∫ z
0
P(t)e Q(t) dt, P and Q
polynomials, then f has no wandering domains. Indeed this follows
from a more general statement.
Theorem (Eremenko-Lyubich, Golberg-Keen 1986): If f is critically
finite then f has no wandering domains.
Remark: The proofs adapted Sullivan’s quasi-conformal strategy.
Wandering domains and Bishop’s folding

Constant limit functions
Theorem (Fatou 1920): Let U a wandering domain of f . All limit
functions of {f n k
| U } are constant (non constant limit functions
correspond to eventually periodic Fatou components).
{f n | U } → ∞ (uniformly tending to infinity)
{f n k
| U } → ∞ and {f m k
| U } → a ∈ J (f ) ⊂ C (oscillating)
There is no {n k } for which {f n k
| U } → ∞ (bounded)
Remark: All previous examples were of the first type.
Example (Eremenko-Lyubich 1987): There exits a transcendental
entire function f which has an oscillating wandering component U
(with infinitely many finite constant limit points).
Remark: As far as I know there are no examples of the third type.
Wandering domains and Bishop’s folding

Composite entire functions and Carleman sets
Theorem (Bergweiler-Wang 1998): Let f , g be entire maps. Then
z ∈ J (f ◦ g) ⇐⇒ g(z) ∈ J (g ◦ f ).
If U 0 ⊂ F(f ◦ g) and V 0 ⊂ F(g ◦ f ) with g(U 0 ) ⊂ V 0 then
U 0 wanders
⇐⇒ V 0 wanders
Main tool: g semi-conjugates f ◦ g with g ◦ f .
Theorem (Singh 2003): There exists U ∈ C and f , g entire maps
such that U is periodic for f , g and g ◦ f but it wanders for f ◦ g.
Remark: Carleman approximation theory.
Problem: It is possible, in general, to relate the existence of
wandering domains for f , g and f ◦ g
Wandering domains and Bishop’s folding

Constant limit functions
Definition: We define
E =
⋃
s∈S(f ) n≥0
⋃
f n (s)
We denote by E ′ the derived set of E, that is, the set of finite limit
points of E. And we denote by E the closure of E.
Theorem (Baker, 1976): Let U a Fatou component of f . Then all
constant limit function of {f n k
| U } belong to E ∪ ∞.
Theorem (BHKMT, 1993): Let U a Fatou component of f . Then
all constant limit function of {f n k
| U } belong to E ′ ∪ ∞.
Corollary: The maps (in class B) f (z) = e z , f (z) = sin(z) ,
z
f (z) =
π2
π 2 −z
sin(z) has no wandering domains.
2
Wandering domains and Bishop’s folding

Eremenko-Lyubich class and wandering domains
Definition: We say that f is in Eremenko-Lyubich class, f ∈ B, if
S(f ) is bounded.
Theorem (Eremenko-Lyubich 1985): If f ∈ B there are no Fatou
components such that {f n | U } converges uniformly to infinity. No
Baker domains and all wandering domains would be, if exist, either
oscillating or bounded. Main result J (f ) = I(f ).
Theorem (Mihaljević-Rempe 2012): If f ∈ B, all singular values
escape uniformly to infinity and f satisfies condition A, then f has
no wandering domains.
Question: Is it possible to erase condition A from the statement
Bishop’s example does not answer to this question since not all
singular values escape to infinity uniformly.
Wandering domains and Bishop’s folding

Bishop’s QC-folding for entire maps (in class B)
The following slices explain Christopher Bishop’s construction of a
wandering domain in class B.
This corresponds to Section 17 on the paper Constructing
entire functions by quasiconformal folding.
I want to thank him for patiently answer my questions while I
was reading the paper.
Of course, possible mistakes or inaccuracies on this exposition,
if any, belongs to me.
Wandering domains and Bishop’s folding

Bishop’s QC-folding for entire maps (in class B)
Assume f is entire with precisely two critical values: {−1, 1}.
Let T = f −1 ([−1, 1]).
Ω j
H +
τ
σ = cosh
C
−1 1
T
C
Then we may write f (z) = cosh (τ(z))
Wandering domains and Bishop’s folding

Bishop’s QC-folding for entire maps (in class B)
Assume f is entire with precisely two critical values: {−1, 1}.
Let T = f −1 ([−1, 1]).
Ω j
H +
τ
σ = cosh
C
−1 1
T
C
f
Main problem: Is it possible to start with an abstract tree T and a
map τ so that f = στ is entire Answer: Formally no, but close to
this.
Wandering domains and Bishop’s folding

Bishop’s QC-folding for entire maps (in class B)
H +
Ω 2
η η = τ
Ω 1
C
σ
−1
1
C
T (r 0 )
η
D
Restriccions:
T should be of uniformly - M - bounded geometry (smooth +
relative size of edges + angle condition).
No common edges for bounded components. Moreover τ
sends vertices of bounded components to roots of unity.
τ-sizes are ≥ π.
The prize: ∃ f = g ◦ φ, g = σ ◦ η q.r and φ K-qc with K = K(M).
Moreover f = σ ◦ τ ◦ φ off T (r 0 ).
Wandering domains and Bishop’s folding

Bishop’s QC-folding for entire maps in class B
Theorem: There exist an entire function f ∈ B, a disc D 0 and an
increasing sequence of integers {n k } ↗ ∞ so that if we set
D n+1 = f (D n ), n ≥ 1, then
(a) The diameter of D n tends to zero (not monotonically),
(b) dist(0, D nk ) ↗ ∞, and
(c) dist(0, D nk +1) < 1.
In particular, f has a wandering domain.
Proof:
D 0 and all its images belong to the Fatou set. Set U 0 the
Fatou component where D 0 lies.
U 0 cannot be (pre)periodic because of (b).
Wandering domains and Bishop’s folding

Bishop’s QC-folding for entire maps in class B
C
π
S k
zk
D k
1
π
2
0 R
1
2
S +
Remark: Symmetry (real and imaginary axis) of the construction.
f (z) = f (z) and f (−z) = −f (z)
Wandering domains and Bishop’s folding

Bishop’s QC-folding for entire maps in class B
π
π
2
C
S k
D k
0 R
1
2
S +
z k
1
τ = λ sinh
η = τ
H +
Remark: The blue and red points correspond to πiZ.
Wandering domains and Bishop’s folding

Bishop’s QC-folding for entire maps in class B
π
C
S k
z D k k
π
2
0 R
1
2 S +
1
τ λ sinh
η = τ
σ(z) = cosh(z)
H +
C
Wandering domains and Bishop’s folding

Bishop’s QC-folding for entire maps in class B
π
π
2
C
S k
z D k k
0 R
1
2 S +
1
τ λ sinh
η = τ
σ(z) = cosh(z)
τ = z − z k
η = τ
d k -th roots of unity
D
H +
σ(z) = ρ(τ(z) d k )
ρ(0) = 1 2 , ρ| ∂D = Id
ρ qc
ρ conformal in D(0, 3/4)
C
Wandering domains and Bishop’s folding

Bishop’s QC-folding for entire maps in class B
σ = cosh
σ = exp
H +
η
π
C
z D k k
π
2
0 R
1
2 S
τ λ sinh
η = τ
S k
g = ση
f = gφ
1
τ = z − z k
η = τ
D
H +
σ σ(z) = ρ(τ d k )
ρ(0) = 1 2 , ρ| ∂D = Id
C
Wandering domains and Bishop’s folding

Bishop’s QC-folding for entire maps in class B
Important remarks about φ:
φ is conformal on S + and 1 4 D k’s.
Wandering domains and Bishop’s folding

Bishop’s QC-folding for entire maps in class B
Important remarks about φ:
φ is conformal on S + and 1 4 D k’s.
φ is uniformly K-quasiconformal for all the values of λ and
d k ’s.
Wandering domains and Bishop’s folding

Bishop’s QC-folding for entire maps in class B
Important remarks about φ:
φ is conformal on S + and 1 4 D k’s.
φ is uniformly K-quasiconformal for all the values of λ and
d k ’s.
The dilatation of φ is supported inside T (r 0 ) and this
neighborhood decays exponentially in |z|.
Wandering domains and Bishop’s folding

Bishop’s QC-folding for entire maps in class B
Important remarks about φ:
φ is conformal on S + and 1 4 D k’s.
φ is uniformly K-quasiconformal for all the values of λ and
d k ’s.
The dilatation of φ is supported inside T (r 0 ) and this
neighborhood decays exponentially in |z|.
Moreover φ is symmetric (1-to-1 on R), φ(0) = 0, φ(∞) = ∞
and
φ(z) = z + a z + O ( |z −2 | )
for some |z| > R (Dyn’kin’s Theorem).
Wandering domains and Bishop’s folding

Bishop’s QC-folding for entire maps in class B
Important remarks about φ:
φ is conformal on S + and 1 4 D k’s.
φ is uniformly K-quasiconformal for all the values of λ and
d k ’s.
The dilatation of φ is supported inside T (r 0 ) and this
neighborhood decays exponentially in |z|.
Moreover φ is symmetric (1-to-1 on R), φ(0) = 0, φ(∞) = ∞
and
φ(z) = z + a z + O ( |z −2 | )
for some |z| > R (Dyn’kin’s Theorem).
φ ′ should be bounded by below from 0.
Estimates get better when increasing the parameters.
Wandering domains and Bishop’s folding

Bishop’s QC-folding for entire maps in class B
C
S k
D k
π
z k
π
2
0 R
1
2
S +
Consequently: f ′ (x) = d dx
cosh (λ sinh(φ(x))) ≥ 16x for λ large
enough. Hence we may choose λ so that x n = f n ( 1
2)
tends to
infinity (exponential speed) as n does.
Wandering domains and Bishop’s folding

Bishop’s QC-folding for entire maps in class B
1/4 ˜D n
U n
· · ·
f n
5
x 0
x n
f n
The nth preimage. The disc U n (1/4 Koebe’s Theorem) has radius
comparable to
( ( )) d 1 −1
r n =
dx f n .
2
Wandering domains and Bishop’s folding

Bishop’s QC-folding for entire maps in class B
D ( x 0 , ( 1 2 )dn )
1/4 ˜D n
1/2 ˜D n
· · ·
x 0
x n
The image by f = g ◦ φ of 1/4 ˜D n .
The map φ sends 1/4 ˜D n inside 1/2 ˜D n (Dyn’kin’s estimates
for n large enough).
The map g(z) = ρ ( (z − ˜z n ) dn) sends D(x 0 , ( 1 2 )dn ).
Wandering domains and Bishop’s folding

Bishop’s QC-folding for entire maps in class B
1/4 ˜D n
f n f
U n
U n+1
D ( x 0 , ( 1 2 )dn )
We want ( 1
2) dn

Bishop’s QC-folding for entire maps in class B
f n ∞
1/4 ˜D n
U n
U n+1
f ∞
D ( x 0 , ( 1 2 )dn )
We want ( 1
2) dn

Bishop’s QC-folding for entire maps in class B
f n ∞
1/4 ˜D n
U n
˜ f ∞
U n+1
f ∞
x 0 = 1 2
We want ( 1
2) dn

Bishop’s QC-folding for entire maps in class B
U n
U n+1
1/4 ˜D n 1/4D ˜ n+1
f ∞ ˜ n
· · ·
f ∞ ˜
f ∞ ˜ n+1
f ∞ ˜
U n+2
x 0 = 1 2 x n x n+1
How he manage to do so...
Wandering domains and Bishop’s folding

Bishop’s QC-folding for entire maps in class B
We change g in the D n ’s to be g = ρ n ρ(τ dn ) so that
ρ n (1/2) = w n+1 the center of U n+1 .
The new dilatation is O(r n+1 ) and it is supported in
O((d n ) −1 ) around the ∂ ˜D n . Notice that the distance from
1/2 to w n+1 is governed by r n+1 .
The correction on φ is close to the identity in the whole plane
but we only need to do so in unit discs centered at the points
{x 0 , . . . x n } so that U n+1 still is mapped to ˜D n+1 (if necessary,
we increase d n ).
We do these slightly modifications to be summable over n.
Wandering domains and Bishop’s folding

The end!!
Thank you!!
Wandering domains and Bishop’s folding