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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
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References:

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