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Dynamic rays of bounded-type transcendental self-maps of the punctured plane. (English) Zbl 1383.37032
The authors study the iteration of holomorphic self-maps of the punctured plane \(\mathbb{C}^* =\mathbb{C}\setminus\{0\}\) having \(0\) and infinity as essential singularities, and they obtain analogues for such functions of several results concerning the dynamics of entire functions.
The authors introduce the class \(\mathcal{B}^*\) of holomorphic self-maps of \(\mathbb{C}^*\) for which the set of singularities of the inverse of \(f\) is bounded away from \(0\) and infinity. This class is the analogue of the Eremenko-Lyubich class \(\mathcal{B}\) of entire functions for which the set of singularities of the inverse of \(f\) is bounded.
The authors prove that for \(f\in\mathcal{B}^*\), the escaping set \(I(f)\), that is the set of all points \(z\in \mathbb{C}^*\) for which the orbit \(\{f^n(z) : n\in\mathbb{N}\}\) has no accumulation points different from \(0\) and infinity, is contained in the Julia set \(J(f)\) of \(f\). They also obtain an analogue of a result of G. Rottenfusser et al. [Ann. Math. (2) 173, No. 1, 77–125 (2011; Zbl 1232.37025)] by proving that if \(f\) is a finite composition of transcendental holomorphic self-maps of \(\mathbb{C}^*\) of finite order, then every point \(z\in I(f)\) can be connected to \(0\) or infinity by a curve \(\gamma\) such that \(f^n|_\gamma\to \{0,\infty\}\) uniformly with respect to the spherical metric.
The paper contains several other interesting results concerning holomorphic self-maps of \(\mathbb{C}^*\).

MSC:
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
37F20 Combinatorics and topology in relation with holomorphic dynamical systems
37F50 Small divisors, rotation domains and linearization in holomorphic dynamics
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