Connectedness properties of the set where the iterates of an entire function are unbounded
Connectedness properties of the set where the iterates of an entire function are unbounded
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DOI：
10.1017/etds.2015.85
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发表时间：
2015-04
影响因子：
0.9
通讯作者：
J. W. Osborne;P. Rippon;G. Stallard
中科院分区：
文献类型：
--
作者：
J. W. Osborne;P. Rippon;G. Stallard
我们研究了整个函数 $f$ 的迭代无界的点集 $I^{\!+\!}(f)$ 的连通性属性。特别是，我们证明只要 $f$ 的最小模数迭代趋于 $\infty$ ，$I^{\!+\!}(f)$ 就会连通。对于一般的超越整个函数 $f$ ，我们证明 $I^{\!+\!}(f)\cup \{\infty \}$ 总是连通的，并且如果 $I^{\!+\!}(f)$ 是断开的，那么它有无数个分量，其中无限多个分量是无界的。
We investigate the connectedness properties of the set $I^{\!+\!}(f)$ of points where the iterates of an entire function $f$ are unbounded. In particular, we show that $I^{\!+\!}(f)$ is connected whenever iterates of the minimum modulus of $f$ tend to $\infty$ . For a general transcendental entire function $f$ , we show that $I^{\!+\!}(f)\cup \{\infty \}$ is always connected and that, if $I^{\!+\!}(f)$ is disconnected, then it has uncountably many components, infinitely many of which are unbounded.