Let f be an entire function and let z0 be an attracting periodic point of f. The immediate attracting basin of z0 is the connected component of the Fatou set that contains z0. Points which tend to infinity under iteration are called escaping. It follows from results of McMullen that if f(z) = λez has an attracting fixed point, then the set of escaping points in the boundary of the attracting basin has Hausdorff dimension 2. In contrast, Bara´nski, Karpi´nska and Zdunik showed for such f that if f has an attracting periodic point of period at least 2, then the set of escaping points in the boundary of the attracting basin has Hausdorff dimension 1. We discuss to which extent these results hold for more general entire functions f. The results are joint work with Jie Ding.
