Speiser class Julia sets with dimension near one
Christopher J. Bishop, Simon Albrecht

TL;DR
This paper constructs entire functions with three singular values whose Julia sets have Hausdorff dimension arbitrarily close to one, filling a gap in the understanding of the possible dimensions of such sets.
Contribution
It provides the first known examples of entire functions with finite singular set and Julia set dimension less than 2, specifically approaching dimension one.
Findings
Constructed entire functions with Julia set dimension near one
Demonstrated existence of functions with finite singular set and low-dimensional Julia sets
Extended understanding of the possible Hausdorff dimensions of Julia sets
Abstract
For any we construct an entire function with three singular values whose Julia set has Hausdorff dimension at most . Stallard proved that the dimension must be strictly larger than 1 whenever has a bounded singular set, but no examples with finite singular set and dimension strictly less than 2 were previously known.
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Taxonomy
TopicsMathematical Dynamics and Fractals · Meromorphic and Entire Functions · Analytic and geometric function theory
