Geometric endomorphisms of the Hesse moduli space of elliptic curves (Q6592702)

This paper deals with geometric endomorphisms of the Hesse moduli space of elliptic curves. The authors consider the geometric map \(\mathcal{C}\), called Cayleyan, associating to a plane cubic \(E\) the adjoint of its dual curve. They show that \(\mathcal{C}\) and the classical hessian map \(\mathcal{H}\) generate a free semigroup. \N\NThey 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 \(\mathcal{W}(\mathcal{H},\mathcal{C})\) generated by \(\mathcal{H},\mathcal{C}\). They point out then how the dynamic behaviours of \(\mathcal{H}\) and \(\mathcal{C}\) differ drastically. \N\NFirstly, concerning the number of real periodic points: for \(\mathcal{H}\) these are infinitely many, for \(\mathcal{C}\) they are just 4. Secondly, the Julia set of \(\mathcal{H}\) is the whole projective line, unlike what happens for all elements of \(\mathcal{W}(\mathcal{H},\mathcal{C})\) which are not iterates of \(\mathcal{H}\). \N\NThis paper is organized as follows: Section 1 is an introduction to the subject, the main results and to the hessian and the Cayley construction. Section 2 deals with Hesse pencil, Hesse group and equations for the hessian and cayleyan maps. Section 3 is devoted to some generalities on periodic points and real periodic points of rational maps. Section 4 deals with semigroup generated by \(\mathcal{H}, \mathcal{C}\) and geometrically special elliptic curves. Section 5 is devoted to the geometry of the map \(\mathcal{C}\) and its real periodic points. Here, the authors begin a qualitative study of the geometry of \(\mathcal{C}\) and of its iterates. It is important to understand the behavior at the fixed points of \(\mathcal{C}\). Section 6 deals with the geometry of the map \(\mathcal{H}\) and its real periodic points. Section 7 is devoted to Julia sets of endomorphisms in the semigroup \(\mathcal{W}(\mathcal{H}, \mathcal{C})\) and directions of further research.