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 \( {\mathfrak {H}}\).