A generalized family of transcendental functions with one dimensional Julia sets

AbstractA generalized family of transcendental (non-polynomial entire) functions is constructed, where the Hausdorff dimension and the packing dimension of the Julia sets are equal to one. Further, there exist multiply connected wandering domains, the dynamics can be completed described, and for any s∈(0,+∞], there is a function taken from this family with the order of growth s. Baker proved that the Hausdorff dimension of a transcendental function is no less than one in 1975, the minimum value was obtained via an elegant construction by Bishop in 2018. The order of growth is zero in Bishop's construction, the family of functions here have arbitrarily positive or even infinite order of growth.Keywords: Fatou setHausdorff dimensionJulia setpacking dimensiontranscendental function2020 Mathematics Subject Classifications: 37F1030D0537F3537C45 AcknowledgementsThe authors appreciate Prof. Christopher Bishop, whose continuous encouragement and support made it possible to finish this work. The authors also appreciate the anonymous reviewers for their helpful comments and suggestions, which improve the manuscript greatly.Disclosure statementNo potential conflict of interest was reported by the author(s).Additional informationFundingThis work was supported by the Shandong Provincial Natural Science Foundation, China [grant number ZR2023MA047].