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In [[mathematics]], the '''Denjoy-Wolff theorem''' is a theorem in [[complex analysis]] and [[dynamical systems]] concerning fixed points and iterations of [[holomorphic mapping]]s of the [[unit disc]] in the [[complex numbers]] into itself, The result was proved independently in 1926 by [[Arnaud Denjoy]] and [[Julius Wolff]]. |
In [[mathematics]], the '''Denjoy-Wolff theorem''' is a theorem in [[complex analysis]] and [[dynamical systems]] concerning fixed points and iterations of [[holomorphic mapping]]s of the [[unit disc]] in the [[complex numbers]] into itself, The result was proved independently in 1926 by [[Arnaud Denjoy]] and [[Julius Wolff]]. |
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==Statement== |
==Statement== |
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'''Theorem''' Let ''D'' be the open unit disc in '''C''' and let ''f'' be a holomorphic function mapping ''D'' into ''D'' which is not an automorphism of ''D'' (i.e. a [[Möbius transformation]]). Then there is a unique point ''z'' in the closure of ''D'' such that the iterates of ''f'' tend to ''z'' uniformly on compact subsets of ''f''. If ''z'' lies in ''D'', it is the unique fixed point of ''f''. The mapping ''f'' leaves invariant [[Poincaré_metric#Metric_and_volume_element_on_the_Poincar.C3.A9_disk|hyperbolic disks]] centered on ''z'' or disks tangent to the unit circle at ''z''. |
'''Theorem.''' Let ''D'' be the open unit disc in '''C''' and let ''f'' be a holomorphic function mapping ''D'' into ''D'' which is not an automorphism of ''D'' (i.e. a [[Möbius transformation]]). Then there is a unique point ''z'' in the closure of ''D'' such that the iterates of ''f'' tend to ''z'' uniformly on compact subsets of ''f''. If ''z'' lies in ''D'', it is the unique fixed point of ''f''. The mapping ''f'' leaves invariant [[Poincaré_metric#Metric_and_volume_element_on_the_Poincar.C3.A9_disk|hyperbolic disks]] centered on ''z'' or disks tangent to the unit circle at ''z''. |
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When the fixed point is at ''z''=0, the hyperbolic disks centred at ''z'' are just the Euclidean disks with centre 0. Otherwise ''f'' can be conjugated by a Möbius transformation so that the fixed point is zero. |
When the fixed point is at ''z''=0, the hyperbolic disks centred at ''z'' are just the Euclidean disks with centre 0. Otherwise ''f'' can be conjugated by a Möbius transformation so that the fixed point is zero. |
Revision as of 12:03, 5 December 2011
In mathematics, the Denjoy-Wolff theorem is a theorem in complex analysis and dynamical systems concerning fixed points and iterations of holomorphic mappings of the unit disc in the complex numbers into itself, The result was proved independently in 1926 by Arnaud Denjoy and Julius Wolff.
Statement
Theorem. Let D be the open unit disc in C and let f be a holomorphic function mapping D into D which is not an automorphism of D (i.e. a Möbius transformation). Then there is a unique point z in the closure of D such that the iterates of f tend to z uniformly on compact subsets of f. If z lies in D, it is the unique fixed point of f. The mapping f leaves invariant hyperbolic disks centered on z or disks tangent to the unit circle at z.
When the fixed point is at z=0, the hyperbolic disks centred at z are just the Euclidean disks with centre 0. Otherwise f can be conjugated by a Möbius transformation so that the fixed point is zero.
References
- Beardon, A. F. (1990), "Iteration of contractions and analytic maps", J. London Math. Soc., 41: 141–150
- Burckel, R. B. (1981), "Iterating analytic self-maps of discs", Amer. Math. Monthly, 88
{{citation}}
: Text "pages 396–407" ignored (help) - Lennart; Gamelin, T. D. W. (1993), Complex dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, ISBN 0-387-97942-5
{{citation}}
: Unknown parameter|First=
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suggested) (help) - Denjoy, A. (1926), "Sur l'itération des fonctions analytiques", C. R. Acad. Scie., 182: 255–257
- Shapiro, J. H. (1993), Composition operators and classical function theory, Universitext: Tracts in Mathematics, Springer-Verlag, ISBN 0-387-94067-7
- Shoikhet, D. (2001), Semigroups in geometrical function theory, Kluwer Academic Publishers, ISBN 0-7923-7111-9
- Steinmetz, Norbert (1993), Rational iteration. Complex analytic dynamical systems, de Gruyter Studies in Mathematics, vol. 16, Walter de Gruyter & Co., ISBN 3-11-013765-8
- Wolff, J. (1926), "Sur l'itération des fonctions holomorphes dans une région, et dont les valeurs appartiennent a cette région", C. R. Acad. Sci., 182: 42–43
- Wolff, J. (1926), "Sur l'itréation des fonctions bornées", C. R. Acad. Sci., 182: 200–201
- Wolff, J. (1926), "Sur une généralisation d'un théorème de Schwarz", C. R. Acad. Sci., 182: 918–920