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Connectedness properties of the set where the iterates of an entire function are unbounded John Osborne (joint work with Phil Rippon and Gwyneth Stallard) Postgraduate Conference in Complex Dynamics 11 - 13 March 2015 The set of points whose

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Connectedness properties of the set where the iterates of an entire function are unbounded

John Osborne

(joint work with Phil Rippon and Gwyneth Stallard)

Postgraduate Conference in Complex Dynamics 11 - 13 March 2015

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The set of points whose orbits are unbounded

f an entire function, f n = f ◦ f ◦ . . . ◦ f (f n(z))n∈N the orbit of z under f K(f) = set of points whose orbits are bounded So K(f)c = set of points whose orbits are unbounded

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The set of points whose orbits are unbounded

f an entire function, f n = f ◦ f ◦ . . . ◦ f (f n(z))n∈N the orbit of z under f K(f) = set of points whose orbits are bounded So K(f)c = set of points whose orbits are unbounded K(f)c ⊃ I(f)

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The set of points whose orbits are unbounded

f an entire function, f n = f ◦ f ◦ . . . ◦ f (f n(z))n∈N the orbit of z under f K(f) = set of points whose orbits are bounded So K(f)c = set of points whose orbits are unbounded K(f)c ⊃ I(f) f a polynomial K(f) = filled Julia set K(f)c = I(f) ⊂ F(f) f transcendental K(f) unbounded K(f)c \ I(f) = ∅

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The set of points whose orbits are unbounded

f an entire function, f n = f ◦ f ◦ . . . ◦ f (f n(z))n∈N the orbit of z under f K(f) = set of points whose orbits are bounded So K(f)c = set of points whose orbits are unbounded K(f)c ⊃ I(f) f a polynomial K(f) = filled Julia set K(f)c = I(f) ⊂ F(f) K(f)c is connected f transcendental K(f) unbounded K(f)c \ I(f) = ∅ When is K(f)c connected?

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Iterating the minimum modulus function

Define m(r) = m(r, f) := min{|f(z)| : |z| = r} mn(r) to be the nth iterate of the function r → m(r). r f

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Iterating the minimum modulus function

Define m(r) = m(r, f) := min{|f(z)| : |z| = r} mn(r) to be the nth iterate of the function r → m(r). r f m(r)

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Iterating the minimum modulus function

Define m(r) = m(r, f) := min{|f(z)| : |z| = r} mn(r) to be the nth iterate of the function r → m(r). r f m(r) f m2(r)

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Iterating the minimum modulus function

Define m(r) = m(r, f) := min{|f(z)| : |z| = r} mn(r) to be the nth iterate of the function r → m(r). r f m(r) f m2(r) f m3(r)

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Iterating the minimum modulus function

Define m(r) = m(r, f) := min{|f(z)| : |z| = r} mn(r) to be the nth iterate of the function r → m(r). r f m(r) f m2(r) f m3(r) For a transcendental entire function (compare the iteration

f M(r)):

∄ R > 0 with m(r) > r ∀ r ≥ R

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Iterating the minimum modulus function

Define m(r) = m(r, f) := min{|f(z)| : |z| = r} mn(r) to be the nth iterate of the function r → m(r). r f m(r) f m2(r) f m3(r) For a transcendental entire function (compare the iteration

f M(r)):

∄ R > 0 with m(r) > r ∀ r ≥ R we can’t always find r > 0 such that mn(r) → ∞ as n → ∞.

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Some functions for which K(f)c is connected

Theorem A Let f be a transcendental entire function for which there exists r > 0 such that mn(r) → ∞ as n → ∞. Then K(f)c is connected.

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Some functions for which K(f)c is connected

Theorem A Let f be a transcendental entire function for which there exists r > 0 such that mn(r) → ∞ as n → ∞. Then K(f)c is connected. Theorem B Let f be a transcendental entire function of order less than 1

2.

Then there exists r > 0 such that mn(r) → ∞ as n → ∞, and therefore K(f)c is connected. Recall that the order ρ of a transcendental entire function is defined as ρ := lim sup

r→∞

log log M(r, f) log r .

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An idea of the proof of Theorem A

Suppose K(f)c is disconnected. Lemma A subset X of C is disconnected if and only if there exists a closed, connected set Γ ⊂ X c such that at least two different components of Γc intersect X.

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An idea of the proof of Theorem A

Suppose K(f)c is disconnected. Lemma A subset X of C is disconnected if and only if there exists a closed, connected set Γ ⊂ X c such that at least two different components of Γc intersect X. mk(r) z0 z1 Γ ⊂ K(f)

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An idea of the proof of Theorem A

Suppose we have a continuum Γ0 ⊂ K(f) such that for some z0 ∈ Γ0, |f n(z0)| < mk(r) for all n ∈ N, and ∃ z1 ∈ Γ0 ∩ {z : |z| = mk(r)}. mk(r) Γ0 z0 z1

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An idea of the proof of Theorem A

Suppose we have a continuum Γ0 ⊂ K(f) such that for some z0 ∈ Γ0, |f n(z0)| < mk(r) for all n ∈ N, and ∃ z1 ∈ Γ0 ∩ {z : |z| = mk(r)}. mk(r) Γ0 z0 z1 mk+N(r) f N(Γ0) f N(z0) f N(z1) mk+N+1(r) N ≥ 1 is the largest integer such that |f N(z1)| ≥ mk+N(r).

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An idea of the proof of Theorem A

Suppose we have a continuum Γ0 ⊂ K(f) such that for some z0 ∈ Γ0, |f n(z0)| < mk(r) for all n ∈ N, and ∃ z1 ∈ Γ0 ∩ {z : |z| = mk(r)}. mk(r) Γ0 z0 z1 mk+N(r) f N(z1) mk+N+1(r) z2 Γ1 Choose Γ1 ⊂ f N(Γ0) so that it contains a point z2 with modulus mk+N(r) but no points

f smaller modulus.

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An idea of the proof of Theorem A

Suppose we have a continuum Γ0 ⊂ K(f) such that for some z0 ∈ Γ0, |f n(z0)| < mk(r) for all n ∈ N, and ∃ z1 ∈ Γ0 ∩ {z : |z| = mk(r)}. mk(r) Γ0 z0 z1 mk+N(r) f N(z1) mk+N+1(r) z2 Γ1 mk+N+L(r) f L(Γ1) f N+L(z1) f L(z2) L ≥ 1 is the largest integer such that |f L(z2)| ≥ mk+N+L(r).

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An idea of the proof of Theorem A

Suppose we have a continuum Γ0 ⊂ K(f) such that for some z0 ∈ Γ0, |f n(z0)| < mk(r) for all n ∈ N, and ∃ z1 ∈ Γ0 ∩ {z : |z| = mk(r)}. mk(r) Γ0 z0 z1 mk+N(r) mk+N+1(r) z2 Γ1 mk+N+L(r) z3 Γ2 Choose Γ2 ⊂ f L(Γ1) so that it contains a point z3 with modulus mk+N+L(r) but no points of smaller modulus.

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An idea of the proof of Theorem A

We have constructed a sequence (Γn) of compact sets such that f kn(Γn) ⊃ Γn+1 for some (kn). mk(r) Γ0 mk+N(r) mk+N+1(r) Γ1 mk+N+L(r) Γ2 f N f L

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An idea of the proof of Theorem A

We have constructed a sequence (Γn) of compact sets such that f kn(Γn) ⊃ Γn+1 for some (kn). mk(r) Γ0 mk+N(r) mk+N+1(r) Γ1 mk+N+L(r) Γ2 f N f L It follows that there is a point in Γ0 with unbounded orbit. #

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Generalising the condition in Theorem A

‘... there exists r > 0 such that mn(r) → ∞ as n → ∞.’ r f m(r) We have: a sequence of nested discs {z : |z| < mn(r)} that fill the plane such that each boundary circle is mapped outside the next disc in the sequence. Can we replace the discs by arbitrary bounded, simply connected domains?

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A more general result

Theorem C Let f be a transcendental entire function, and (Dn)n∈N be a sequence of bounded, simply connected domains such that (a) f(∂Dn) surrounds Dn+1, for n ∈ N, and (b) every disc centred at 0 is contained in Dn for sufficiently large n. Then K(f)c is connected.

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A more general result

Theorem C Let f be a transcendental entire function, and (Dn)n∈N be a sequence of bounded, simply connected domains such that (a) f(∂Dn) surrounds Dn+1, for n ∈ N, and (b) every disc centred at 0 is contained in Dn for sufficiently large n. Then K(f)c is connected. Is this really more general than Theorem A?

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A more general result

Theorem C Let f be a transcendental entire function, and (Dn)n∈N be a sequence of bounded, simply connected domains such that (a) f(∂Dn) surrounds Dn+1, for n ∈ N, and (b) every disc centred at 0 is contained in Dn for sufficiently large n. Then K(f)c is connected. Is this really more general than Theorem A? Example: Let f(z) = −10ze−z − 1

2z.

Note that m(r) ∼ 1

2r as r → ∞, so Theorem A does not hold.

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Example: f(z) = −10ze−z − 1

2z

4nπ nπi −nπi