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Dynamics of tangent. (English) Zbl 0662.30019
Dynamical systems, Proc. Spec. Year, College Park/Maryland, Lect. Notes Math. 1342, 105-111 (1988).
[For the entire collection see Zbl 0653.00011.]
The idea of Julia set \(J(f)\) may be extended from the case of rational or entire maps \(z\to f(z)\) to the case when \(f\) is meromorphic. The authors discuss the example \(f(z)=\lambda \tan z\), where \(\lambda\) is a parameter. If \(\lambda >1\) then \(J=\mathbb{R}\). If \(\lambda\) is real and \(0<| \lambda | <1\) then J is a Cantor subset of \(\mathbb{R}\) and the action of f on J is conjugate to a ‘shift on infinitely many symbols’. If \(\lambda\in\mathbb{C}\), \(0<| \lambda | <1\), then \(J(\lambda)\) is a Cantor set and there is a quasiconformal map of the plane which conjugates \(z\to \lambda \tan z\) on \(J(\lambda)\) to \(z\to \tan z\) on \(J()\). Neither \(J=\mathbb{R}\) nor \(J\) a Cantor set is possible for transcendental entire functions.
Reviewer: I.N.Baker

MSC:
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
37B99 Topological dynamics