p-Fractals and power series–I. Some 2 variable results

u₁,…,u_r are in kx₁,…,x_s with k and deg(u₁,…,u_r) finite. Intending applications to Hilbert–Kunz theory, we code the numbers into a function φ_u, which empirically satisfies many functional equations related to “magnification by p,” where p=chark. p-fractals, introduced here, formalize these ideas.In the first interesting case (r=3, s=2), the φ_u are p-fractals. Our proof uses functions φ_I attached to ideals I and square-free elements h of A=kx,y. The finiteness of the set of ideal classes in and the existence of “magnification maps” on this set show the φ_I to be p-fractals.We describe further functional equations coming from a theory of reflection maps on ideal classes, and the paper concludes with examples and open questions.