## Valiron, Nevanlinna and Picard exceptional sets of iterations of rational functions.(English)Zbl 1094.30030

Suppose $$f$$ is a rational function on the Riemann sphere. Denote the $$k$$ times iteration of $$f$$ by $$f^k$$ for $$k\in\mathbb{N}$$. The complex number $$a\in\mathbb{C}\cup\{\infty\}$$ is said to be a Picard exceptional value of $$\{f^k\}_{k\in\mathbb{N}}$$ if the number of the elements of $$\bigcup_{k\in\mathbb{N}} f^{-k}(a)$$ is finite. The Picard exceptional set $$E(\{f^k\})$$ is defined by that of all such points. The Valiron and Nevanlinna defects are defined as $\delta_V(a; \{f^k\})= \limsup_{k\to\infty} {m(a, f^k)\over d_k}\quad\text{and}\quad \delta_N(a; \{f^k\})= \liminf{m(a, f^k)\over d_k},$ where $$m(a,f^k)$$ is the mean proximity of $$f^k$$ with respect to $$a$$. The Valiron and Nevanlinna exceptional sets $$E_V(\{f^k\})$$ and $$E_N(\{f^k\})$$ are defined by those of all points with nonzero Valiron and Nevanlinna defects, respectively. The author proves that $$E(\{f^k\})= E_V(\{f^k\})= E_N(\{f^k\})$$ for $$k> 1$$.

### MSC:

 30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable 39B32 Functional equations for complex functions 37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
Full Text:

### References:

 [1] H. Brolin, Invariant sets under iteration of rational functions, Ark. Mat. 6 (1965), 103-144. · Zbl 0127.03401 · doi:10.1007/BF02591353 [2] A. È. Erëmenko and M. L. Sodin, Teor. Funktsiĭ Funktsional. Anal. i Prilozhen. (1990), No. 53 18-25; translation in J. Soviet Math. 58 (1992), no. 6, 504-509. [3] A. Freire, A. Lopes and R. Mané, An invariant measure for rational maps, Bol. Soc. Brasil. Mat. 14 (1983), no. 1, 45-62. · Zbl 0568.58027 · doi:10.1007/BF02584744 [4] K. Ishizaki and N. Yanagihara, Deficiency for meromorphic solutions of Schröder equations, Complex Var. Theory Appl. 49 (2004), no. 7-9, 539-548. · Zbl 1099.30012 · doi:10.1080/02781070410001731701 [5] K. Ishizaki and N. Yanagihara, Borel and Julia directions of meromorphic Schröder functions, Math. Proc. Camb. Phil. Soc. (To appear). · Zbl 1076.30034 [6] M. Ju. Ljubich, Entropy properties of rational endomorphisms of the Riemann sphere, Ergodic Theory Dynam. Systems 3 (1983), no. 3, 351-385. · Zbl 0537.58035 · doi:10.1017/S0143385700002030 [7] 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. · Zbl 0979.37001 [8] R. Nevanlinna, Analytic functions , Translated from the second German edition by Phillip Emig. Die Grundlehren der mathematischen Wissenschaften, 162, Springer, New York, 1970. · Zbl 0199.12501 [9] 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. · Zbl 0793.30026 [10] N. Yanagihara, Exceptional values for meromorphic solutions of some difference equations, J. Math. Soc. Japan 34 (1982), no. 3, 489-499. · Zbl 0478.30020 · doi:10.2969/jmsj/03430489
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.