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• W1548993673 abstract "Given a compact holomorphic fixed-point-free self-map, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=f> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding=application/x-tex>f</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, of the open unit ball of a Hilbert space, we show that the sequence of iterates, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis f Superscript n Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:msup> <mml:mi>f</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>(f^n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, converges locally uniformly to a constant map <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=xi> <mml:semantics> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:annotation encoding=application/x-tex>xi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-vertical-bar xi double-vertical-bar equals 1> <mml:semantics> <mml:mrow> <mml:mo fence=false stretchy=false>‖<!-- ‖ --></mml:mo> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:mo fence=false stretchy=false>‖<!-- ‖ --></mml:mo> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>Vert xi Vert = 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This extends results of Denjoy (1926), Wolff (1926), Hervé (1963) and MacCluer (1983). The result is false without the compactness assumption, nor is it true for all open balls of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper J Superscript asterisk> <mml:semantics> <mml:msup> <mml:mi>J</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding=application/x-tex>J^{*}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-algebras." @default.
• W1548993673 created "2016-06-24" @default.
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• W1548993673 date "1997-01-01" @default.
• W1548993673 modified "2023-10-14" @default.
• W1548993673 title "Iteration of compact holomorphic maps on a Hilbert ball" @default.
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• W1548993673 doi "https://doi.org/10.1090/s0002-9939-97-03761-1" @default.
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