fatou components and singularities of meromorphic functions...the half plane frez < 2gis...

Fatou components and singularitiesof meromorphic functions

Bogus lawa Karpinska

Faculty of Mathematics and Information ScienceWarsaw University of Technology

On geometric complexity of Julia sets II2020

joint work with Krzysztof Baranski, Nuria Fagella and Xavier Jarque

B.Karpinska Fatou components and singularities 1 / 20

Transcendental dynamics

f : C→ C transcendental entire or meromorphic function

∞ is an essential singularity of f

The successive iterates of f are not defined at the poles of f and their preimages.



Transcendental maps may have Fatou components that are neither basinsof attraction nor rotation domains.

A periodic Fatou component U, ofperiod p, is called Baker domain ifthere exists z0 ∈ ∂U such that

f np(z)→ z0

for z ∈ U but f p(z0) is not defined.



Fatou’s example:Let f (z) = z + 1 + e−z .

The right half plane {z : Rez > 0} iscontained in invariant Baker domain.

Fatou components which are specific for transcendental meromorphic maps:

Baker domains

wandering domains

Fatou component U is a wandering domain if f n(U) ∩ f m(U) = ∅ for all n 6= m.


Singular valuesSingular set

S(f ) = {critical and asymptotic values of f }

A point a ∈ C is an asymptotic value if there exists a curve γ : (0, 1)→ Csuch that γ(t)→∞ as t → 1− and f (γ(t))→ a.

Postsingular set

P(f ) =⋃

s∈S(f )

∞⋃n=0

f n(s)

Singular values play a special role.

Any basin of attraction of an attracting or parabolic cycle containsa singular value.

If U is a Siegel disc or Herman ring then ∂U ⊂ P(f ).

The relation between singular values Baker domains or wandering domains is lessclear.


Singular values

If S(f ) is finite (class S) then f has no Baker domains nor wanderingdomains [Eremenko-Lyubich’92], [Baker-Kotus-Lu’ 92].

If S(f ) is bounded (class B) and f is entire then f has no Baker domainsnor escaping wandering domains [Eremenko-Lyubich’ 92].

In general, meromorphic map with bounded singular set can have Bakerdomains [Baker-Kotus-Lu’ 92], but they have no escaping wanderingdomains [Rippon-Stallard’ 99].


Example (Bergweiler ’ 95)

The mapf (z) = 2− log 2 + 2z − ez

has Baker domain U such that

dist(U,P(f )) > c > 0.

Notice that f is a lift of entire map h(z) = 12z

2e2−z , which has 2 fixed points: 0and 2, both superattracting. The singular set consists of critical values

S(f ) = {zk = log 2 + 2πki , k ∈ Z},

P(f ) = {z2k : k ∈ Z}.

The half plane {Rez < −2} is contained in an invariant Baker domain U.


Example (Bergweiler ’ 95)Let U0 be the basin of superattracting fixed point log 2.Consider its copies Uk = U0 + 2kπi , Uk ∈ F(f ).Then U ∩ Uk = ∅.


Singular values and Baker domains

Theorem (Bergweiler ’95)

Let f be transcendental entire, let U be its invariant Baker domain s.t.U ∩ S(f ) = ∅. Then there exists a sequence pn ∈ P(f ) ∩ C such that:

(a) |pn| → ∞

(b)∣∣∣ pn+1

pn

∣∣∣→ 1

(c) dist(pn,U) = o(|pn|)

Theorem (Mihaljevic-Brandt, Rempe-Gillen ’13)

The above theorem holds for transcendental meromorphic maps with condition∣∣∣∣pn+1

pn

∣∣∣∣→ 1 replaced by supn≥1

∣∣∣∣pn+1

pn

∣∣∣∣ <∞.B.Karpinska Fatou components and singularities 9 / 20

Singular values and Baker domains

Theorem 1 (Baranski, Fagella, Jargue, K.)

Let U be a periodic Baker domain for a meromorphic map f s.t f pnU →∞. IfC \ U has an unbounded component then there exists a sequence pn ∈ P(f ) ∩ Csuch that:

(a) |pn| → ∞

(b)∣∣∣ pn+1

pn

∣∣∣→ 1


The assumption that C \ U has an unbounded component is always satisfied forNewton’s method for entire maps.


Singular values and wandering domainsQuestion: Let U be a wandering domain for a meromorphic map f s.t.Um ∩ S(f ) = ∅. Is there some relation between ∂Un and S(f )?

Theorem 2 (Baranski, Fagella, Jargue, K.)

Let f be a transcendental meromorphic function and let U be its Fatoucomponent (wandering domain). Let Un be a component containing f n(U).Then for every z ∈ U there exists a sequence pn ∈ P(f ) such that

dist(pn,Un)

dist(f n(z), ∂Un)→ 0.

In particular, if the diameter of Un is uniformly bounded, then dist(pn,Un)→ 0.


Topologically hyperbolic maps

Definitionf transcendental meromorphic is topologically hyperbolic if

dist(P(f ),J (f ) ∩ C) > 0.

Such maps can have sequences of postsingular points accumulating at infinity.

Corollary

Let f be topologically hyperbolic meromorphic map, U-its Fatou component,Un-Fatou component containing f n(U). Suppose that Un ∩ P(f ) = ∅ for n > 0.Then for every compact set K ⊂ U and every r > 0 there exists n0 such that forevery z ∈ K

D(f n(z), r) ⊂ Un for n ≥ n0.


No wandering domains

Newton’s method for the function

h(z) = aez + bz + c

has no wandering domains1 (for many parameters a, b, c).

Nh(z) = z − h(z)

h′(z)=

z − 1− αe−z

1 + βe−z

has exactly one asymptotic value u = −α/β. If α, β ∈ C, β 6= 0 and

infn≥0

dist (Nnh (u), J(Nh)) > 0

then Nh is topologically hyperbolic. This is the case for ab > 0 orab < 0, c/b ≥ 1− ln(− b

a ).

1proved earlier by Bergweiler-Terglane and Kriete using quasiconformal deformationstechnique


Figure: Dynamical plane for the function f (z) = ez (z−1)ez+1 , Newton’s method for

h(z) = z + ez .


Main ingredient of the proofs

LemmaLet f be a transcendental meromorphic function, U-its Fatou component,Un- Fatou component containing f n(U). Then for every compact set K ∈ U, forevery ε, for every sufficiently large n for every z ∈ K and for every curve γ joiningf n(z) to the boundary of Un such that l(γ) � dist(f n(z), ∂Un) there exists apoint pn ∈ P(f ) in εl(γ)-neighbourhood of γ.


Proof

Assume that there exist a set K ⊂ U, sequences nj →∞ and zj ∈ K such that:

Let gj be a branch of f −nj on D1 sending f nj to zj . The distortion of gj isbounded on the chain of discs joining D1 and D2 by a constant independent on j .


Idea of the proof of Theorem 1

Our aim is to prove that if U is a periodic Baker domain for a meromorphic mapf s.t f pnU →∞, C \ U has an unbounded component then there exists a sequencepn ∈ P(f ) ∩ C such that:

(a) |pn| → ∞

(b)∣∣∣ pn+1

pn

∣∣∣→ 1


Take z ∈ U, let K be a curve joining z to f p(z) in U,

Γ =∞⋃n=0

f pm(K )

and apply the lemma for ε = 1/k and arc of circles intersecting U.


Idea of the proof of Theorem 1


Thank you for your attention!


fatou components and singularities of meromorphic functions...the half plane frez < 2gis...

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