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A generalized family of transcendental (non-polynomial entire) functions isconstructed, where the Hausdorff dimension and the packing dimension of theJulia sets are equal to one. Further, there exist multiply connected wanderingdomains, the dynamics can be completed described, and for any$s\in(0,+\infty]$, there is a function taken from this family with the order ofgrowth $s$. Baker proved that the Hausdorff dimension of the transcendentalfunction is no less than one in 1975, the minimum value was obtained via anelegant construction by Bishop in 2018. The order of growth is zero in Bishop'sconstruction, the family of functions here have arbitrarily positive or eveninfinite order of growth.

A generalized family of transcendental functions with one dimensional Julia sets