The 3 rd International Conference on Applied Mathematics and Informatics Dedicated to memory of Alexander Vasil’ev Value regions of univalent self-maps with two boundary fixed points Pavel Gumenyuk (University of Stavanger, Norway) San Andrés – COLOMBIA, 26/11 - 1/12/2017 1/7
Universitetet i Introduction Stavanger Consider a holomorphic self-map ϕ : D → D := { z ∈ C : | z | < 1 } . Definition A point σ ∈ ∂ D is called a boundary regular fixed point (BRFP) of ϕ if ∠ lim z → σ ϕ ( z ) exists and equals σ , and if the angular derivative of ϕ at σ , ϕ ( z ) − σ ϕ ′ ( σ ) := ∠ lim is finite. z − σ z → σ ϕ ′ ( σ ) > 0. REMARK: for every BRFP σ , Denjoy – Wolff point � If ϕ � id D has a (unique) fixed point τ ∈ D , then we call τ the Denjoy – Wolff point (DW-point). � If ϕ has no fixed point in D , then by the Denjoy – Wolff Theorem, ∃ ! BRFP τ ∈ ∂ D with ϕ ′ ( τ ) � 1, called the DW-point of ϕ . 2/7
Universitetet i Introduction 2 Stavanger Dynamical meaning of the DW-point � If ϕ � id D and it is not an elliptic automorphism of D , then τ is the attracting fixed point, i.e. ϕ ◦ n := ϕ ◦ . . . ◦ ϕ ( n times) → τ as n → + ∞ . � All BRFPs σ ∈ ∂ D \ { τ } are repelling, i.e. ϕ ′ ( σ ) > 1. We mostly will consider univalent ( = holomorphic + injective) ϕ : D → D with given BRFPs. ✒ H. Unkelbach, 1938, 1940 ✒ C.C. Cowen and Chr. Pommerenke, 1982 ✒ Chr. Pommerenke and A. Vasil’ev, 2001, 2002 ✒ A. Vasil’ev, 2002 ✒ M.D. Contreras, S. Díaz-Madrigal, and A. Vasil’ev, 2007 ✒ J.M. Anderson and A. Vasil’ev, 2008 ✒ A. Frolova, M. Levenshtein, D. Shoikhet, and A. Vasil’ev, 2014 ✒ V. Goryainov, 1991, 2015, 2017 3/7
Universitetet i Main results Stavanger Conf. map ℓ : D → S := { ζ : − π/ 2 < I m ζ < π/ 2 } ; z �→ log 1 + z 1 − z ; ± 1 �→ ±∞ . Joint work with Prof. Dmitri Prokhorov: Theorem ( to appear in Ann. Acad. Sci. Fenn. Math. 43 (2018)) Fix T > 0 , z 0 ∈ D . The value region V ( z 0 , T ) of ϕ �→ ϕ ( z 0 ) over the class of all univalent self-maps ϕ : D → D having: (ii) a BRFP σ = − 1 with ϕ ′ ( σ ) = e T , (i) the DW-point τ = 1 and is a closed Jordan domain with the boundary point z 0 excluded. More precisely, ℓ ( V ( z 0 , T ) ∪ { z 0 } ) = ✔ � � � � x + iy ∈ S : y 1 T � y � y 2 � x − T T , 2 − Re ℓ ( z 0 ) � � F T ( y ) , � � where y 1 T , y 2 T , and F T ( · ) are given explicitly. ✔ Furthermore, every boundary point ω ∈ ∂ V ( z 0 , T ) \ { z 0 } is delivered by a unique ϕ = ϕ ω , which is a hyperbolic automorphism if ω = ℓ − 1 ( ℓ ( z 0 ) + T ) and a parabolic one-slit map otherwise. 4/7
Universitetet i Main results 2 Stavanger On the left: z 0 := i / 2, T ∈ { log 2 , log 4 , log 6 } . On the right: z 0 := 0, T := log 6, (and the value region without taking into account univalence). i i i � 2 i � 2 � 1 � 0.5 1 � 1 0.5 1 � i � i • ℓ − 1 ( ℓ ( z 0 ) + T ) ∂ D ∂ V ( z 0 , T ) ◦ z 0 5/7
Universitetet i Main results 3 — Methods Stavanger Corollary ( Theorem 6 in [Frolova, Levenshtein, Shoikhet, Vasil’ev, 2014]) For any univalent self-map ϕ : D → D with BRFPs at ± 1 , � ϕ ′ ( − 1 ) ϕ ′ ( 1 ) � Φ( I m ℓ ( z 0 ) , I m ℓ ( ϕ ( z 0 ))) , (1) � 1 + sin α � 1 + sin β , 1 − sin α where Φ( α, β ) := max . Estimate (1) is sharp. 1 − sin β Methods ✔ Cowen – Pommerenke inequalities for univalent self-maps: Grunsky-type inequalities; ✔ A. Vasil’ev: (a specific form of the) Extremal Length Method; ✔ V. Goryanov: a Loewner-type Parametric Representation (for the case of τ = 0 and one BRFP). 6/7
Universitetet i Parametric Method Stavanger Loewner-type Parametric Representation Embed ϕ into a family ϕ t : D → D , t ∈ [ 0 , 1 ] , ϕ 0 = id D , ϕ 1 = ϕ , s.t. � �� � d ϕ t ( z ) / d t = τ − ϕ t ( z ) 1 − τϕ t ( z ) p ( ϕ t ( z ) , t ) , (LK) where Re p � 0 plus some other conditions. PGum, 2017, 2018 : Loewner-type Parametric Representation for ϕ ’s with any finite number of BRFPs and any position of the DW-point. This gives a new proof of a classical Cowen – Pommerenke inequality [ joint work in progress with M.D. Contreras and S. Díaz-Madrigal ]: The value region of log ϕ ′ ( 0 ) over all univalent ϕ : D → D , ϕ ( 0 ) = 0, with BRFPs σ j , . . . , σ n with prescribed values of ϕ ′ ( σ j ) ’s is described n 1 � 1 1 � � � by the inequality Re − log ϕ ′ ( σ j ) . (CP) log ϕ ′ ( 0 ) 2 j = 1 7/7
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