Source: Proc. Japan Acad. Ser. A Math. Sci.
Volume 81, Number 2
For every rational function of degree more than one, there
exists a transcendental meromorphic solution of the Schröder
equation. By Yanagihara and Eremenko-Sodin, it is known that
the Valiron, Nevanlinna and Picard exceptional sets of this
solution are all same.
As an analogue of this result,
we show that all the Valiron, Nevanlinna and Picard exceptional
sets of iterations of a rational function of degree more than
one are also same. As a corollary, the equidistribution theorem
in complex dynamics follows.
H. Brolin, Invariant sets under iteration of rational functions, Ark. Mat. 6 (1965), 103--144.
Mathematical Reviews (MathSciNet): MR194595
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Mathematical Reviews (MathSciNet): MR736568
K. Ishizaki and N. Yanagihara, Deficiency for meromorphic solutions of Schröder equations, Complex Var. Theory Appl. 49 (2004), no. 7-9, 539--548.
K. Ishizaki and N. Yanagihara, Borel and Julia directions of meromorphic Schröder functions, Math. Proc. Camb. Phil. Soc. (To appear).
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Mathematical Reviews (MathSciNet): MR741393
S. Morosawa, Y. Nishimura, M. Taniguchi and T. Ueda, Holomorphic dynamics, Translated from the 1995 Japanese original and revised by the authors, Cambridge Studies in Advanced Mathematics, 66, Cambridge Univ. Press, Cambridge, 2000.
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Mathematical Reviews (MathSciNet): MR279280
M. Sodin, Value distribution of sequences of rational functions, in Entire and subharmonic functions, Adv. Soviet Math., 11, Amer. Math. Soc., Providence, RI, 1992, pp. 7--20.
N. Yanagihara, Exceptional values for meromorphic solutions of some difference equations, J. Math. Soc. Japan 34 (1982), no. 3, 489--499.
Mathematical Reviews (MathSciNet): MR659617