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Examples of entire functions with pathological dynamics. (English) Zbl 0601.30033
We use the approximation theory in the complex domain to construct several examples of entire functions with pathological behaviour of iterates. The main example is an entire function f such that the family of its iterates is normal in a domain D and this family has an infinite set of constant limit functions in D. Another example shows that an entire function can have an finite-dimensional family of invariant measurable fields of directions on its Julia set. In this example the iterates tend to infinity on a subset of the Julia set having positive area.

MSC:
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
30E10 Approximation in the complex plane
37-XX Dynamical systems and ergodic theory
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