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

u1,…,ur are in k x1,…,xs with k and deg(u1,…,ur) finite. Intending applications to Hilbert–Kunz theory, we code the numbers into a function φu, 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 φu are p-fractals. Our proof uses functions φI attached to ideals I and square-free elements h of A=k x,y . The finiteness of the set of ideal classes in and the existence of “magnification maps” on this set show the φI 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.