This work is motivated by the paper Annular itineraries for entire - - PowerPoint PPT Presentation
Annular Itineraries for C David Mart-Pete Dept. of Mathematics and Statistics The Open University supervised by Phil Rippon and Gwyneth Stallard Winter School on Kleinian Groups and Transcendental Dynamics Universitt Bremen
David Martí-Pete
The Open University — supervised by Phil Rippon and Gwyneth Stallard —
Winter School on Kleinian Groups and Transcendental Dynamics Universität Bremen April 10, 2014
This work is motivated by the paper Annular itineraries for entire functions by Phil Rippon and Gwyneth Stallard (January 2013).
and annuli covering results
Let f : S ⊆ C → S be holomorphic. By Montel’s thm., if J (f ) = ∅ then:
◮ S =
C = C ∪ {∞}, the Riemann sphere (rational functions);
◮ S = C, the complex plane (transcendental entire functions); ◮ S = C∗ = C \ {0}, the punctured plane.
Holomorphic self-maps of C∗ were first studied in 1953 by H. Rådström.
1969.
Let f : S ⊆ C → S be holomorphic. By Montel’s thm., if J (f ) = ∅ then:
◮ S =
C = C ∪ {∞}, the Riemann sphere (rational functions);
◮ S = C, the complex plane (transcendental entire functions); ◮ S = C∗ = C \ {0}, the punctured plane.
Holomorphic self-maps of C∗ were first studied in 1953 by H. Rådström.
Theorem (Bhattacharyya 1969)
Every transcendental holomorphic function f : C∗ → C∗ is of the form f (z) = zn exp
1969.
Let f : S ⊆ C → S be holomorphic. By Montel’s thm., if J (f ) = ∅ then:
◮ S =
C = C ∪ {∞}, the Riemann sphere (rational functions);
◮ S = C, the complex plane (transcendental entire functions); ◮ S = C∗ = C \ {0}, the punctured plane.
Holomorphic self-maps of C∗ were first studied in 1953 by H. Rådström.
Theorem (Bhattacharyya 1969)
Every transcendental holomorphic function f : C∗ → C∗ is of the form f (z) = zn exp
Later results are due to: L. Keen, J. Kotus, P. M. Makienko, A. N. Mukha- medshin, L. Fang, W. Bergweiler, I. N. Baker, P. Domínguez Soto, etc.
1969.
The complexification of the Arnol’d standard family Fαβ(θ) = θ + α + β sin(θ) (mod 2π), θ ∈ R are transcendental hol. self-maps of C∗: Fαβ(w) = weiαeβ(w−1/w)/2. α = 3.1, β = 0.8. α = 3.1, β = 5.
1–31.
Relation with the lift F : C → C (i.e. exp ◦F ≡ f ◦ exp):
◮ exp−1 J (f ) = J (F). [Ber95]
About the Julia set:
◮ J (f ) has no compact component; [BD98] ◮ J (f ) has either one or infinitely many components. [BD98]
About the Fatou set: if f is not a Möbius transformation,
◮ the Fatou components are simply or doubly-connected and there is
at most one doubly connected component (if f (z) ≡ kzn); [Kee88]
◮ if A is doubly-connected comp. which is relatively compact in C∗,
then A is either a Herman ring, pre-periodic, or wandering. [BD98]
Theory Appl. 37 (1998), no. 1-4, 67–91.
The escaping set was firstly studied for transcendental entire functions by
I(f ) := {z ∈ C : |f n(z)| → ∞}.
Theorem (Eremenko 1989, Eremenko & Lyubich 1992)
Let f be a transcendental entire function. Then,
◮ I(f ) ∩ J (f ) = ∅; ◮ J (f ) = ∂I(f ); ◮ all the components of I(f ) are unbounded; ◮ if f ∈ B, then I(f ) ⊆ J (f ).
Banach Center Publ. 23 (1989), 339-345.
If f : C∗ → C∗ with 0, ∞ essential singularities, I(f ) := {z ∈ C∗ :
which contains I0(f ) := {z ∈ C∗ : |f n(z)| → 0}, I∞(f ) := {z ∈ C∗ : |f n(z)| → ∞} and also for (sn) ∈ {−1, +1}N Is(f ) :=
|f n+ℓ(z)| > 1 if sn = +1, |f n+ℓ(z)| < 1 if sn = −1
Theorem (Fang 1998, Baker, Domínguez & Herring 2001)
Let f be a transcendental self-map of C∗, then
◮ J (f ) ∩ I0(f ) = ∅, J (f ) ∩ I∞(f ) = ∅; ◮ J (f ) = ∂I0(f ) = ∂I∞(f ). I.N. Baker, P. Domínguez and M.E. Herring, Dynamics of functions meromorphic outside a small set, Ergodic Theory and Dynamical Systems 21 (2001), no. 3, 647–672. Fang L., On the Iteration of Holomorphic Self-Maps of C∗, Acta Mathematica Sinica, New Series 14 (1998), no. 1, 139–144.
f (z) = z + 1 + e−z: Cantor bouquet (blue) & Baker domain (red).
Let f be a holomorphic self-map of C∗ and R > 0, the functions M(R) = max
|z|=R |f (z)| < +∞,
m(R) = min
|z|=R |f (z)| > 0
are continuous and unimodal.
Let f be a holomorphic self-map of C∗ and R > 0, the functions M(R) = max
|z|=R |f (z)| < +∞,
m(R) = min
|z|=R |f (z)| > 0
are continuous and unimodal. Main properties: i)
log M(R) log R
→ ∞ as R → ∞; ii) log M(R) is a convex function of log R; iii) ∃R0 > 0 such that M(Rk) M(R)k for every R R0 and k > 1; iv) if k > 1, M(kR)
M(R) → ∞ as R → ∞.
Let f be a holomorphic self-map of C∗ and R > 0, the functions M(R) = max
|z|=R |f (z)| < +∞,
m(R) = min
|z|=R |f (z)| > 0
are continuous and unimodal. Main properties: i)
log M(R) log R
→ ∞ as R → ∞; ii) log M(R) is a convex function of log R; iii) ∃R0 > 0 such that M(Rk) M(R)k for every R R0 and k > 1; iv) if k > 1, M(kR)
M(R) → ∞ as R → ∞.
The analogous properties hold at 0, and similarly for m(R).
Let R+ be big enough such that for all R > R+, M(R) > R, and let R− be small enough such that for all R < R−, m(R) < R. Define A0 := A(R−, R+) = {z ∈ C∗ : R− < |z| < R+} and, for every n > 0, An := A(Mn−1(R+), Mn(R+)), A−n := A(mn(R−), mn−1(R−)).
The annular itinerary of z ∈ C∗ is a0a1a2 . . ., with an ∈ Z, and f n(z) ∈ Aan, for all n 0.
The annular itinerary of z ∈ C∗ is a0a1a2 . . ., with an ∈ Z, and f n(z) ∈ Aan, for all n 0. What kind of sequence can be realized as an itinerary? By construction, an+1 an + 1 if an > 0, an+1 an − 1 if an < 0. because of the Maximum Modulus Principle.
We will use the following well-known result:
Lemma
Let Cm, m 0, be compact sets in C and f continuous such that f (Cm) ⊇ Cm+1, for m 0. Then ∃ξ such that f m(ξ) ∈ Cm, for m 0. We are going to construct a sequence of annuli Bn ⊂ An such that
◮ f (Bn) ⊇ Bn+1 if n > 0; ◮ f (Bn) ⊇ Bn−1 if n < 0.
Theorem (Bergweiler, Rippon & Stallard 2011)
There exists an absolute constant δ > 0 such that if f : A(R, R′) → C∗ is analytic, where R′ > R, then for all z1, z2 ∈ A(R, R′) such that ρA(R,R′)(z1, z2) < δ and |f (z2)| 2|f (z1)|, we have f (A(R, R′)) ⊃ A(|f (z1)|, |f (z2)|).
functions, to appear in Proc. London Math. Soc.; arXiv:1109.1794
Let µ(R) := εM(R) for ε > 0. For large enough values of R, µn(R) Mn(εR) for n ∈ N.
(3) 105 (2012) 787–820.
Let µ(R) := εM(R) for ε > 0. For large enough values of R, µn(R) Mn(εR) for n ∈ N. We have the following situation: for n > 0, Mn−1(R+) < ε2M(µn−1(R+)) < µn(R+) < M(µn−1(R+)) < Mn(R+).
(3) 105 (2012) 787–820.
Let µ(R) := εM(R) for ε > 0. For large enough values of R, µn(R) Mn(εR) for n ∈ N. We have the following situation: for n > 0, Mn−1(R+) < ε2M(µn−1(R+)) < µn(R+) < M(µn−1(R+)) < Mn(R+). We are going to take, for n > 0, Bn := A(ε2M(µn−1(R+)), M(µn−1(R+))) ⋐ An and z1, z2 such that |z1| = |z2| = µn(R+) such that |z1| = m(µn(R+)), |z2| = M(µn(R+)).
(3) 105 (2012) 787–820.
Let µ(R) := εM(R) for ε > 0. For large enough values of R, µn(R) Mn(εR) for n ∈ N. We have the following situation: for n > 0, Mn−1(R+) < ε2M(µn−1(R+)) < µn(R+) < M(µn−1(R+)) < Mn(R+). We are going to take, for n > 0, Bn := A(ε2M(µn−1(R+)), M(µn−1(R+))) ⋐ An and z1, z2 such that |z1| = |z2| = µn(R+) such that |z1| = m(µn(R+)), |z2| = M(µn(R+)). Observe that making ε small enough we can ensure that ρAn(z1, z2) < δ and for R+ big enough the condition |f (z2)| 2|f (z1)| is trivial.
(3) 105 (2012) 787–820.
Observe that we can choose R− and R+ respectively small and big enough such that
◮ if n > 0: f (Bn) ⊇ Bk for all 0 < k n + 1; ◮ if n < 0: f (Bn) ⊇ Bk for all n − 1 k < 0.
Observe that we can choose R− and R+ respectively small and big enough such that
◮ if n > 0: f (Bn) ⊇ Bk for all 0 < k n + 1; ◮ if n < 0: f (Bn) ⊇ Bk for all n − 1 k < 0.
We introduce the following functions
◮ for n > 0, σ+(n) = min{k ∈ Z : f (Bn) ⊇ Bk} 0; ◮ for n < 0, σ−(n) = max{k ∈ Z : f (Bn) ⊇ Bk} 0;
from which we only know that they are resp. non-increasing and non- decreasing.
Theorem (M)
Let f : C∗ → C∗ be holomorphic with 0, ∞ essential singularities. Given a partition {An} defined as above with R+, 1/R− big enough, we can construct points with the following annular itineraries:
◮ fast escaping points to either 0, ∞ or both; ◮ periodic itineraries, in given consecutive annuli in the
neighbourhood of 0 or ∞, or jumping from one to the other;
◮ uncountably many bounded itineraries; ◮ uncountably many unbounded non-escaping itineraries; ◮ arbitrarily slowly escaping itineraries.
We say that a point z ∈ C∗ is fast escaping if there is ℓ ∈ N and a sequence of positive real numbers (Rn)n starting with R0 > 0 sufficiently large/small and given by Rn+1 = M(Rn) or Rn+1 = m(Rn) for n > 0, such that
◮ |f n+ℓ(z)| Rn if Rn+1 = M(Rn); ◮ |f n+ℓ(z)| Rn if Rn+1 = m(Rn).
We write z ∈ A(f ). Remark: fast escaping points with this definition are indeed escaping!
If f is a transcendental entire function, Bergweiler and Hinkkanen showed that A(f ) = ∅ in the original paper from 1999. This construction proves that, with this definition, A(f ) = ∅ for holomor- phic self-maps with two essential singularities as well. Moreover, it can be adapted to show that if s ∈ {−1, +1}N is a sequence describing which of the essential singularities the escaping points are visiting (sn = −1 if f n(z) is in a neighbourhood of 0 and sn = +1 otherwise) and As(f ) := {z ∈ A(f ) : ∃ℓ ∈ N s.t. f ℓ(z) has escaping itinerary s} then for every s ∈ {−1, +1}N ∅ = As(f ) ⊆ I(f ). Observe that sn = sign(an).
In 2005, Rippon and Stallard proved that every component of A(f ) is unbounded and hence I(f ) has at least one unbounded component.
Theorem (M)
Let f be a holomorphic self-map of C∗ with 0, ∞ essential singularities. The connected components of A(f ) are unbounded. Hence, A(f ) contains uncountably many disjoint unbounded continua attached to each essential singularity.
(2005), no. 4, 1119–1126.