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Semigroups of Holomorphic Self-Maps of the Disk Potential Theory

Semigroups of Holomorphic Functions and Hyperbolic Capacity

Maria Kourou

Aristotle University of Thessaloniki

Postgraduate Conference in Complex Dynamics, London 2019

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Semigroups of Holomorphic Self-Maps of the Disk Potential Theory

Joint work with: D. Betsakos, G. Kelgiannis and St. Pouliasis

This research is carried out / funded in the context of the project Condenser Capacity and Holomorphic Functions (MIS 5004684) under the call for proposals Supporting researchers with emphasis on new researchers (EDULLL 34). The project is co-financed by Greece and the European Union (European Social Fund- ESF) by the Operational Programme Human Resources Development, Education and Lifelong Learning 2014-2020. Maria Kourou Semigroups of Holomorphic Functions and Hyperbolic Capacity 2 / 21

Semigroups of Holomorphic Self-Maps of the Disk Potential Theory Characterization of Semigroups Orbits Koenigs function

Outline

1

Semigroups of Holomorphic Self-Maps of the Disk Characterization of Semigroups Orbits Koenigs function

2

Potential Theory Hyperbolic Capacity Outcomes Equilibrium Measures

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Semigroups of Holomorphic Self-Maps of the Disk Potential Theory Characterization of Semigroups Orbits Koenigs function

A one-parameter semigroup is a family (φt)t≥0 of holomorphic self-maps in D, where (i) φ0(z) = z; (ii) φt+s(z) = φt (φs(z)), for every t, s ≥ 0 and z ∈ D; (iii) φt(z) t→0+ − − − → z, uniformly on every compact subset of D.

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Semigroups of Holomorphic Self-Maps of the Disk Potential Theory Characterization of Semigroups Orbits Koenigs function

Characterization of Semigroups

Definition There exists a unique point τ ∈ D such that for every z ∈ D, lim

t→+∞ φt(z) = τ.

This point is the Denjoy-Wolff point of the semigroup. τ ∈ D and φt / ∈ EAut(D) for any t ≥ 0 ⇒ elliptic semigroup. τ = 1 and ∠ lim

z→1 φt(z) = 1.

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Semigroups of Holomorphic Self-Maps of the Disk Potential Theory Characterization of Semigroups Orbits Koenigs function

The angular derivative φ′

t(1) = ∠ lim z→1

φt(z) − 1 z − 1 ≤ 1. φ′

t(1) < 1 ⇒ hyperbolic semigroup

φ′

t(1) = 1 ⇒ parabolic semigroup

parabolic of zero hyperbolic step if dD(φt(z), φt+s(z))

t→+∞

− − − − → 0, ∀s > 0, z ∈ D

• therwise, parabolic of positive hyperbolic step

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Semigroups of Holomorphic Self-Maps of the Disk Potential Theory Characterization of Semigroups Orbits Koenigs function

Trajectory of a point

The curve γz : [0, +∞) → D with γz(t) = φt(z). is the trajectory of z ∈ D and lim

t→+∞ γz(t) =

lim

t→+∞ φt(z) = τ,

∀z ∈ D

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Semigroups of Holomorphic Self-Maps of the Disk Potential Theory Characterization of Semigroups Orbits Koenigs function

Koenigs function

For every semigroup (φt) with D-W point 1, there exists a conformal mapping h of D, with h(0) = 0, such that h(φt(z)) = h(z) + t, ∀z ∈ D, t ≥ 0.

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Semigroups of Holomorphic Self-Maps of the Disk Potential Theory Characterization of Semigroups Orbits Koenigs function

The simply connected domain Ω = h(D) is convex in the horizontal direction, as {w + s : s > 0} ⊂ Ω, ∀w ∈ Ω. Hyperbolic semigroup ⇔ Ω ⊂ S Parabolic of positive h. s. ⇔ Ω ⊂ H The base space Ω⋆ is the smallest horizontal domain including Ω and the triple (Ω⋆, h, φt) is called holomorphic model.

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Semigroups of Holomorphic Self-Maps of the Disk Potential Theory Characterization of Semigroups Orbits Koenigs function

Question

Suppose K ⊂ D is compact of positive logarithmic capacity. The γK(t) :=

• z∈K

γz(t) =

• z∈K

φt(z) = φt(K) is the trajectory of K. Its image under h is h(φt(K)) = h(K) + t. Intuition: As t increases, φt(K) move to ∂ D and it is getting ‘smaller’. Question How does φt(K) contract to a point?

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Semigroups of Holomorphic Self-Maps of the Disk Potential Theory Hyperbolic Capacity Outcomes Equilibrium Measures

Outline

1

Semigroups of Holomorphic Self-Maps of the Disk Characterization of Semigroups Orbits Koenigs function

2

Potential Theory Hyperbolic Capacity Outcomes Equilibrium Measures

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Semigroups of Holomorphic Self-Maps of the Disk Potential Theory Hyperbolic Capacity Outcomes Equilibrium Measures

Hyperbolic Capacity

Suppose K ⊂ D is compact. Its hyperbolic n-th diameter is dD

n,h(K) =

sup

w1,...,wn∈K

• 1≤µ<ν≤n
• wµ − wν

1 − wµwν

• 2

n(n−1)

The hyperbolic capacity of K is caph K = lim

n→+∞ dD n,h(K)

and it is a conformally invariant quantity.

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Semigroups of Holomorphic Self-Maps of the Disk Potential Theory Hyperbolic Capacity Outcomes Equilibrium Measures

Theorem The caph φt(K) is a strictly decreasing function of t ≥ 0, unless φt0 is an automorphism of D for some t0 > 0. In this case, caph φt(K) = caph K, for every t ≥ 0.

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Semigroups of Holomorphic Self-Maps of the Disk Potential Theory Hyperbolic Capacity Outcomes Equilibrium Measures

Results

Let (φt)t≥0 be a hyperbolic or a parabolic semigroup of positive h. s. Theorem (Betsakos, Kelgiannis, K., Pouliasis, 2018) The lim

t→+∞ caph φt(K) = caphΩ⋆ h(K),

where caphΩ⋆ is the hyperbolic capacity with respect to the hyperbolic geometry of Ω⋆.

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Semigroups of Holomorphic Self-Maps of the Disk Potential Theory Hyperbolic Capacity Outcomes Equilibrium Measures

Zero Hyperbolic Step

Let (φt)t≥0 be a parabolic semigroup of zero h.s. Theorem (Betsakos, Kelgiannis, K., Pouliasis, 2018) The limit lim

t→+∞ caph φt(K) = 0.

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Semigroups of Holomorphic Self-Maps of the Disk Potential Theory Hyperbolic Capacity Outcomes Equilibrium Measures

Let K ⊂ D compact. Set gD the Green function of D. The hyperbolic capacity of K is equal to caph K =

• inf

µ

• gD(x, y)dµ(x)dµ(y)

−1 , where µ is a Borel measure with compact support K and µ(E) = 1. The infimum is attained for a Borel measure µ, which is called equilibrium measure of K.

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Semigroups of Holomorphic Self-Maps of the Disk Potential Theory Hyperbolic Capacity Outcomes Equilibrium Measures

Set µt the eq. m. of φt(K) and νt = h⋆µt. Then νt is the eq. m.

• f h(K) + t with respect to Ω.

Hence, νt compose a family of Borel measures. Question What can we say about the convergence of the family (νt)t?

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Semigroups of Holomorphic Self-Maps of the Disk Potential Theory Hyperbolic Capacity Outcomes Equilibrium Measures

For a measure µ with compact support in D, denote its norm by µ2 :=

• gD(x, y)dµ(x)dµ(y)

(Green energy). There are three types of convergence for a sequence of measures, strong, weak and vague. We are interested in the following. Definition The sequence (µn) converges strongly to a measure µ if lim

n→∞ µn − µ = 0.

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Semigroups of Holomorphic Self-Maps of the Disk Potential Theory Hyperbolic Capacity Outcomes Equilibrium Measures

Suppose that (φt) is either a hyperbolic or a parabolic semigroup

• f positive h.s.

Theorem (K., 2018) The family (νt)t≥0 converges strongly to ν⋆, as t → +∞, where ν⋆ is the equilibrium measure of h(K) with respect to Ω⋆. Namely, it holds that lim

t→∞

• gD(x, y)d(νt − ν⋆)(x)d(νt − ν⋆)(y) = 0.

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Semigroups of Holomorphic Self-Maps of the Disk Potential Theory Hyperbolic Capacity Outcomes Equilibrium Measures

Similar results have been acquired for several other geometric and potential theoretic quantities. Some examples are Harmonic measure Hyperbolic area Green potential Condenser capacity Hyperbolic n-th diameter

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Semigroups of Holomorphic Self-Maps of the Disk Potential Theory Hyperbolic Capacity Outcomes Equilibrium Measures

Thank you for your attention!

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