The Symmetrized Bidisc and Lempert's Theorem

Costara, Constantin

doi:10.1112/s0024609304003200

Cited by 91 publications

(58 citation statements)

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“…The rich and surprising geometry of this domain and its higher-dimensional analogues has subsequently been elaborated by many authors (for example, [12,18,20,25,26]).…”

Section: Introductionmentioning

confidence: 99%

Extremal holomorphic maps and the symmetrized bidisc

Agler

Lykova

Young

2012

Proceedings of the London Mathematical Society

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We introduce the class of n‐extremal holomorphic maps, a class that generalizes both finite Blaschke products and complex geodesics, and apply the notion to the finite interpolation problem for analytic functions from the open unit disc into the symmetrized bidisc Γ. We show that a well‐known necessary condition for the solvability of such an interpolation problem is not sufficient whenever the number of interpolation nodes is 3 or greater. We introduce a sequence 𝒞ν, where ν⩾0, of necessary conditions for solvability, prove that they are of strictly increasing strength and show that 𝒞n−3 is insufficient for the solvability of an n‐point problem for n⩾3. We propose the conjecture that condition 𝒞n−2 is necessary and sufficient for the solvability of an n‐point interpolation problem for Γ and we explore the implications of this conjecture. We introduce a classification of rational Γ‐inner functions, that is, analytic functions from the disc into Γ whose radial limits at almost all points on the unit circle lie in the distinguished boundary of Γ. The classes are related to n‐extremality and the conditions 𝒞ν; we prove numerous strict inclusions between the classes.

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“…The rich and surprising geometry of this domain and its higher-dimensional analogues has subsequently been elaborated by many authors (for example, [12,18,20,25,26]).…”

Section: Introductionmentioning

confidence: 99%

Extremal holomorphic maps and the symmetrized bidisc

Agler

Lykova

Young

2012

Proceedings of the London Mathematical Society

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“…[2,3,4], [7], [16]). Description of automorphisms in G 2 was given in [12], description of proper mappings in G 2 was given in [9].…”

Section: Introductionmentioning

confidence: 99%

Geometry of the symmetrized polydisc

2005

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“…G has proved to have a rich and explicit function theory, as developed and generalised in [7,11,13] and papers by several other authors. In the function theory and geometry of G much depends on the striking properties of certain rational functions of 3 variables:…”

Section: Introductionmentioning

confidence: 99%

The Magic Functions and Automorphisms of a Domain

Agler

Young

2008

Complex anal.oper. theory

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Abstract. We introduce the notion of magic functions of a general domain in C d and show that the set of magic functions of a given domain is an intrinsic complex-geometric object. We determine the set of magic functions of the symmetrised bidisc G and thereby find all automorphisms of G and a formula for the Carathéodory distance on G.Mathematics Subject Classification (2000). 32M05, 32F45, 32F32, 46A55.

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The Symmetrized Bidisc and Lempert's Theorem

Cited by 91 publications

References 2 publications

Extremal holomorphic maps and the symmetrized bidisc

Extremal holomorphic maps and the symmetrized bidisc

Geometry of the symmetrized polydisc

The Magic Functions and Automorphisms of a Domain

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