The dynamics of the Hesse derivative on the j-invariant

The j-invariant of a cubic curve is an isomorphism invariant parameterized by the moduli space of elliptic curves. The Hesse derivative of a curve $V(f)$ given by the homogeneous polynomial f is V(H(f)) where H(f) is a the determinant of the Hesse matrix of f. In this paper, we compute the j-invariant of the Hesse derivative of a cubic curve C in terms of the j-invariant of C, getting a rational function on the Riemann sphere. We then analyze the dynamics of this rational function, and investigate when a cubic curve is isomorphic to its n-fold Hesse derivative.