Fixed point set for entire functions

The fixed point theorem for whole functions is a theorem of complex analysis , which goes back to a work by the French mathematician Pierre Fatou from 1926. It was rediscovered by the American mathematician Paul C. Rosenbloom in 1948 and subsequently further generalized.

The theorem is a consequence of Picard's Little Theorem .

Formulation of the sentence

"The chained function   always has a fixed point for a whole function , unless     a translation is   with   ." ${\displaystyle f\colon \,\mathbb {C} \to \mathbb {C} }$   ${\displaystyle {f\circ f}\colon \,\mathbb {C} \to \mathbb {C} }$ ${\displaystyle f}$   ${\displaystyle z\mapsto f(z)=z+z_{0}}$ ${\displaystyle z_{0}\in \mathbb {C} \setminus \{0\}}$

Demarcation

In general, entire functions     themselves are “free of fixed points”, ie they have no fixed points. ${\displaystyle f\colon \,\mathbb {C} \to \mathbb {C} }$

A simple example of this is provided by the function   , which is definitely "free of fixed points" because the complex exponential function has   no zeros . ${\displaystyle z\mapsto f(z)={z+\exp(z)}}$   ${\displaystyle \exp }$

literature

Original work

• Detlef Bargmann, Walter Bergweiler: Periodic points and normal families . In: Proc. Amer. Math. Soc . tape 129 , 2001, p. 2881-2888 ( MR1840089 ).

• Pierre Fatou: Sur l'itération des fonctions transcendantes Entières . In: Acta Math . tape 47 , 1926, pp. 337-370 ( MR1555220 ).
• Paul C. Rosenbloom : L'itération des fonctions entières . In: CR Acad. Sci. Paris . tape 227 , 1948, pp. 382-383 ( MR0026691 ).
• PC Rosenbloom: The fix points of entire functions . In: Medd. Lunds Univ. Mat. Sem. Tome Supplémentaire . 1952, p. 186-192 ( MR0051916 ).

Monographs

• Robert B. Burckel: An Introduction to Classical Complex Analysis . Birkhäuser Verlag, Basel [u. a.] 1979, ISBN 3-7643-0989-X .
• Reinhold Remmert , Georg Schumacher: Function theory 2 (=  Springer textbook - basic knowledge of mathematics ). 3rd, revised edition. Springer Verlag, Berlin [u. a.] 2007, ISBN 978-3-540-40432-3 .

Web link

• Link to the further article by Bargmann and Bergweiler (PDF; 180 kB)

Individual evidence

1. ↑ Fatou: Sur l'itération des fonctions transcendantes Entières . In: Acta Math. Band 47 , p. 345 .
2. ↑ Burckel: pp. 433, 458, 559.
3. ↑ Rosenbloom: L'iteration des fonctions entières . In: CR Acad. Sci. Paris . tape 227 , p. 382-383 .
4. ↑ Rosenbloom: The fix points of entire functions . In: Medd. Lunds Univ. Mat. Sem. Tome Supplémentaire . 1952, p. 186 ff .
5. ↑ Burckel: p. 433.
6. ↑ Remmert, Schumacher: pp. 233-234.