A transcendental Julia set of dimension 1

We construct a non-polynomial entire function whose Julia set has finite 1-dimensional spherical measure, and hence Hausdorff dimension 1. In 1975, Baker proved the dimension of such a Julia set must be at least 1, but whether this minimum could be attained has remained open until now. Our example also has packing dimension 1, and is the first transcendental Julia set known to have packing dimension strictly less than 2. It is also the first example with a multiply connected wandering domain where the dynamics can be completely described.