Semigroups of Holomorphic Functions and Hyperbolic Capacity Maria - PowerPoint PPT Presentation

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 Maria Kourou Semigroups of Holomorphic Functions and Hyperbolic Capacity 1 / 21

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

Characterization of Semigroups Semigroups of Holomorphic Self-Maps of the Disk Orbits Potential Theory Koenigs function Outline Semigroups of Holomorphic Self-Maps of the Disk 1 Characterization of Semigroups Orbits Koenigs function Potential Theory 2 Hyperbolic Capacity Outcomes Equilibrium Measures Maria Kourou Semigroups of Holomorphic Functions and Hyperbolic Capacity 3 / 21

Characterization of Semigroups Semigroups of Holomorphic Self-Maps of the Disk Orbits Potential Theory Koenigs function A one-parameter semigroup is a family ( φ t ) t ≥ 0 of holomorphic self-maps in D , where (i) φ 0 ( z ) = z ; (ii) φ t + s ( z ) = φ t ( φ s ( z )), for every t , s ≥ 0 and z ∈ D ; (iii) φ t ( z ) t → 0 + − − − → z , uniformly on every compact subset of D . Maria Kourou Semigroups of Holomorphic Functions and Hyperbolic Capacity 4 / 21

Characterization of Semigroups Semigroups of Holomorphic Self-Maps of the Disk Orbits Potential Theory Koenigs function Characterization of Semigroups Definition There exists a unique point τ ∈ D such that for every z ∈ D , t → + ∞ φ t ( z ) = τ. lim This point is the Denjoy-Wolff point of the semigroup. τ ∈ D and φ t / ∈ EAut( D ) for any t ≥ 0 ⇒ elliptic semigroup . τ = 1 and ∠ lim z → 1 φ t ( z ) = 1 . Maria Kourou Semigroups of Holomorphic Functions and Hyperbolic Capacity 5 / 21

Characterization of Semigroups Semigroups of Holomorphic Self-Maps of the Disk Orbits Potential Theory Koenigs function The angular derivative φ t ( z ) − 1 φ ′ t (1) = ∠ lim ≤ 1 . z − 1 z → 1 φ ′ t (1) < 1 ⇒ hyperbolic semigroup φ ′ t (1) = 1 ⇒ parabolic semigroup parabolic of zero hyperbolic step if t → + ∞ d D ( φ t ( z ) , φ t + s ( z )) − − − − → 0 , ∀ s > 0 , z ∈ D otherwise, parabolic of positive hyperbolic step Maria Kourou Semigroups of Holomorphic Functions and Hyperbolic Capacity 6 / 21

Characterization of Semigroups Semigroups of Holomorphic Self-Maps of the Disk Orbits Potential Theory Koenigs function Trajectory of a point The curve γ z : [0 , + ∞ ) → D with γ z ( t ) = φ t ( z ) . is the trajectory of z ∈ D and t → + ∞ γ z ( t ) = lim t → + ∞ φ t ( z ) = τ, lim ∀ z ∈ D Maria Kourou Semigroups of Holomorphic Functions and Hyperbolic Capacity 7 / 21

Characterization of Semigroups Semigroups of Holomorphic Self-Maps of the Disk Orbits Potential Theory Koenigs function Koenigs function For every semigroup ( φ t ) with D-W point 1, there exists a conformal mapping h of D , with h (0) = 0, such that h ( φ t ( z )) = h ( z ) + t , ∀ z ∈ D , t ≥ 0 . Maria Kourou Semigroups of Holomorphic Functions and Hyperbolic Capacity 8 / 21

Characterization of Semigroups Semigroups of Holomorphic Self-Maps of the Disk Orbits Potential Theory 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 , φ t ) is called holomorphic model . Maria Kourou Semigroups of Holomorphic Functions and Hyperbolic Capacity 9 / 21

Characterization of Semigroups Semigroups of Holomorphic Self-Maps of the Disk Orbits Potential Theory Koenigs function Question Suppose K ⊂ D is compact of positive logarithmic capacity. The � � γ K ( t ) := γ z ( t ) = φ t ( z ) = φ t ( K ) z ∈ K z ∈ K is the trajectory of K . Its image under h is h ( φ t ( K )) = h ( K ) + t . Intuition: As t increases, φ t ( K ) move to ∂ D and it is getting ‘smaller’. Question How does φ t ( K ) contract to a point? Maria Kourou Semigroups of Holomorphic Functions and Hyperbolic Capacity 10 / 21

Hyperbolic Capacity Semigroups of Holomorphic Self-Maps of the Disk Outcomes Potential Theory Equilibrium Measures Outline Semigroups of Holomorphic Self-Maps of the Disk 1 Characterization of Semigroups Orbits Koenigs function Potential Theory 2 Hyperbolic Capacity Outcomes Equilibrium Measures Maria Kourou Semigroups of Holomorphic Functions and Hyperbolic Capacity 11 / 21

Hyperbolic Capacity Semigroups of Holomorphic Self-Maps of the Disk Outcomes Potential Theory Equilibrium Measures Hyperbolic Capacity Suppose K ⊂ D is compact. Its hyperbolic n-th diameter is 2 � � w µ − w ν n ( n − 1) � d D � � n , h ( K ) = sup � � 1 − w µ w ν w 1 ,..., w n ∈ K � � 1 ≤ µ<ν ≤ n The hyperbolic capacity of K is n → + ∞ d D caph K = lim n , h ( K ) and it is a conformally invariant quantity. Maria Kourou Semigroups of Holomorphic Functions and Hyperbolic Capacity 12 / 21

Hyperbolic Capacity Semigroups of Holomorphic Self-Maps of the Disk Outcomes Potential Theory Equilibrium Measures Theorem The caph φ t ( K ) is a strictly decreasing function of t ≥ 0 , unless φ t 0 is an automorphism of D for some t 0 > 0 . In this case, caph φ t ( K ) = caph K, for every t ≥ 0 . Maria Kourou Semigroups of Holomorphic Functions and Hyperbolic Capacity 13 / 21

Hyperbolic Capacity Semigroups of Holomorphic Self-Maps of the Disk Outcomes Potential Theory Equilibrium Measures Results Let ( φ t ) t ≥ 0 be a hyperbolic or a parabolic semigroup of positive h. s. Theorem (Betsakos, Kelgiannis, K., Pouliasis, 2018) The t → + ∞ caph φ t ( K ) = caph Ω ⋆ h ( K ) , lim where caph Ω ⋆ is the hyperbolic capacity with respect to the hyperbolic geometry of Ω ⋆ . Maria Kourou Semigroups of Holomorphic Functions and Hyperbolic Capacity 14 / 21

Hyperbolic Capacity Semigroups of Holomorphic Self-Maps of the Disk Outcomes Potential Theory Equilibrium Measures Zero Hyperbolic Step Let ( φ t ) t ≥ 0 be a parabolic semigroup of zero h.s. Theorem (Betsakos, Kelgiannis, K., Pouliasis, 2018) The limit t → + ∞ caph φ t ( K ) = 0 . lim Maria Kourou Semigroups of Holomorphic Functions and Hyperbolic Capacity 15 / 21

Hyperbolic Capacity Semigroups of Holomorphic Self-Maps of the Disk Outcomes Potential Theory Equilibrium Measures Let K ⊂ D compact. Set g D the Green function of D . The hyperbolic capacity of K is equal to � − 1 � �� caph K = inf g D ( x , y ) d µ ( x ) d µ ( y ) , µ 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 . Maria Kourou Semigroups of Holomorphic Functions and Hyperbolic Capacity 16 / 21

Hyperbolic Capacity Semigroups of Holomorphic Self-Maps of the Disk Outcomes Potential Theory Equilibrium Measures Set µ t the eq. m. of φ t ( K ) and ν t = h ⋆ µ t . Then ν t is the eq. m. of h ( K ) + t with respect to Ω. Hence, ν t compose a family of Borel measures. Question What can we say about the convergence of the family ( ν t ) t ? Maria Kourou Semigroups of Holomorphic Functions and Hyperbolic Capacity 17 / 21

Hyperbolic Capacity Semigroups of Holomorphic Self-Maps of the Disk Outcomes Potential Theory Equilibrium Measures For a measure µ with compact support in D , denote its norm by �� � µ � 2 := g D ( x , y ) d µ ( x ) d µ ( 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 ( µ n ) converges strongly to a measure µ if n →∞ � µ n − µ � = 0 . lim Maria Kourou Semigroups of Holomorphic Functions and Hyperbolic Capacity 18 / 21

Hyperbolic Capacity Semigroups of Holomorphic Self-Maps of the Disk Outcomes Potential Theory Equilibrium Measures Suppose that ( φ t ) is either a hyperbolic or a parabolic semigroup of positive h.s. Theorem (K., 2018) The family ( ν t ) 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 �� g D ( x , y ) d ( ν t − ν ⋆ )( x ) d ( ν t − ν ⋆ )( y ) = 0 . lim t →∞ Maria Kourou Semigroups of Holomorphic Functions and Hyperbolic Capacity 19 / 21

Hyperbolic Capacity Semigroups of Holomorphic Self-Maps of the Disk Outcomes Potential Theory 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 Maria Kourou Semigroups of Holomorphic Functions and Hyperbolic Capacity 20 / 21