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Totally disconnected Julia set for different classes of meromorphic functions. (English) Zbl 1333.37023
Let \(X\) be a Riemann surface. A map \(f\) is of “finite type” if \(f:D\rightarrow X\) is analytic where \(D\subset X\) is an open set, \(f\) is not locally constant, \(f\) has only finitely singular values and every isolated singularity of \(f\) is essential. This notion was introduced by A. Epstein [Towers of finite type complex analytic maps. New York: City University of New York, Graduate Center (PhD Thesis) (1993)].
In the short paper under review, the authors study the dynamics of finite type maps. They describe the global structure of the Julia set of \(f\). More precisely, they prove that if \(f\) is a generic finite type maps having an attracting fixed point which basin contains all singular values of \(f\), then the Julia set \(J_f\) of \(f\) is totally disconnected.
Their proof follows closely the one of I. N. Baker et al. [Ergodic Theory Dyn. Syst. 21, No. 3, 647–672 (2001; Zbl 0990.37033)] in the case of meromorphic maps: they first prove that the attracting basin is totally invariant. In a second time, they prove that the complement of an open neighborhood of the attracting fixed point is a finite union of disks. Finally, they conclude by a classical argument. One of the key ingredients is the classification of Fatou components for finite type maps established by Epstein.
They also provide some examples.

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
37F50 Small divisors, rotation domains and linearization in holomorphic dynamics
Full Text: DOI
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