1
$\begingroup$

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?

$\endgroup$
1
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ Thank you for answering my question. Just to make sure I understand, could you provide a few concrete examples? $\endgroup$ – D.S. Lipham Feb 9 at 16:28
  • $\begingroup$ 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 Feb 10 at 1:47

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.