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I am looking for examples of transcendental entire functions $f:\mathbb C\to \mathbb C$ such that the set of non-escaping points in the Julia set of $f$ is not totally disconnected. I denote this set $J_r(f)=\{z\in J(f):f^n(z)\not\to\infty\}$ because in some interesting cases it is the same as the "radial Julia set'' of $f$.

For two types functions, including the exponential family of $\exp(z)-2$, I have recently shown that $J_r(f)$ is not only totally disconnected, but is zero-dimensional in the topological sense: my paper. The Julia sets of these functions have a relatively simple "Cantor bouquet'' structure which was used in the proofs.

Question. Can $J_r(f)$ contain non-degenerate or unbounded connected sets?

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If $f$ has order $<1/2$ then there is a sequence $r_k\to\infty$ with the property that $$\min_{|z|=r_k}|f(z)|>r_k.$$ Restricting $f$ on $\{ z:|z|<r_k\}$ we obtain a polynomial-like map in the sense of Douady and Hubbard. If $J_k$ is the Julia set of this map, then evidently $J_k\subset J(f)$, and the points of $J_k$ are not escaping. Now, if $f$ has an attracting cycle, then for large $k$, $J_k$ contains the boundary of the attraction domain of this cycle, which is a continuum. Thus $J(f)$ contains a continuum consisting of non-escaping points.

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  • $\begingroup$ Thank you for answering my question. Just to make sure I understand, could you provide a few concrete examples? $\endgroup$
    – D.S. Lipham
    Commented Feb 9, 2020 at 16:28
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    $\begingroup$ Examples of functions of order less than $1/2$? Plenty. Take an appropriate power series, for example. To arrange an attracting cycle, multply on $z^2$. $\endgroup$
    – Alexandre Eremenko
    Commented Feb 10, 2020 at 1:47

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