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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
##### Keywords:
meromorphic map; shift map; Julia set