Geometric Endomorphisms of the Hesse moduli space of elliptic curves

We consider the geometric map $\mathfrak{C}$, called Cayleyan, associating to a plane cubic $E$ the adjoint of its dual curve. We show that $\mathfrak{C}$ and the classical Hessian map $\mathfrak{H}$ generate a free semigroup. We begin the investigation of the geometry and dynamics of these maps, and of the geometrically special elliptic curves: these are the elliptic curves isomorphic to cubics in the Hesse pencil which are fixed by some endomorphism belonging to the semigroup $\mathcal{W}(\mathfrak{H},\mathfrak{C})$ generated by $\mathfrak{H},\mathfrak{C}$. We point out then how the dynamic behaviours of $\mathfrak{H}$ and $\mathfrak{C}$ differ drastically. Firstly, concerning the number of real periodic points: for $\mathfrak{H}$ these are infinitely many, for $\mathfrak{C}$ they are just $4$. Secondly, the Julia set of $\mathfrak{H}$ is the whole projective line, unlike what happens for all elements of $\mathcal{W}(\mathfrak{H},\mathfrak{C})$ which are not iterates of $\mathfrak{H}$.