The Hausdorff dimension of Julia sets of entire functions. II. (English) Zbl 0852.30018
Let \(f\) be a transcendental entire function and \(J(f)\) the Julia set of \(f\). In an earlier paper [Ergodic Theory Dyn. Syst. 11, No. 4, 769-777 (1991; Zbl 0738.58027)], the author proved that the Hausdorff dimension \(d(f)\) of \(J(f)\) satisfies \(1\leq d(f)\leq 2\), while for any \(\delta> 0\) there are examples of \(f\) such that \(d(f)< 1+ \delta\). It is an open question whether \(d(f)= 1\) is possible.
In the present work, the author considers the class \(B\) of transcendental entire functions \(f\) such that the singularities of \(f^{- 1}\) lie in a bounded set. She shows that for any \(f\) in \(B\), \(d(f)> 1\). The method involves an elaborate construction of certain measures on \(J(f)\) similar to those used by Sullivan for Julia sets of rational functions. An example from the class \(B\) is used to throw light on the difficulties of extending the ideas of the earlier example with \(d(f)< 1+ \delta\) construct an example with \(d= 1\).
In the present work, the author considers the class \(B\) of transcendental entire functions \(f\) such that the singularities of \(f^{- 1}\) lie in a bounded set. She shows that for any \(f\) in \(B\), \(d(f)> 1\). The method involves an elaborate construction of certain measures on \(J(f)\) similar to those used by Sullivan for Julia sets of rational functions. An example from the class \(B\) is used to throw light on the difficulties of extending the ideas of the earlier example with \(d(f)< 1+ \delta\) construct an example with \(d= 1\).
Reviewer: I.N.Baker (London)
MSC:
| 30D05 | Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable |
Citations:
Zbl 0738.58027References:
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| [5] | Sullivan, Geometric dynamics, Lecture Notes in Math. 1007 pp 725– (1983) · doi:10.1007/BFb0061443 |
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| [11] | Stallard, Ergod. Th. and Dynam. Sys. 11 pp 769– (1991) |
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