On the Hilbert-Kunz function of zD−p4(x,y)

k is a field of characteristic p, λ∈k, λ≠0 or 1, and g(x,y,z)=zD−xy(x+y)(x+λy) with p≡±1(D). Under a certain hypothesis, Z∗(l) on l=⌊pD⌋ (which has been verified for l⩽11) we calculate en(g)=deg(xpn,ypn,zpn,g) for all n. The technique is based on the p-fractals of Teixeira and the author; the very concrete results are stated in terms of a family of dynamical systems parametrized by λ.