Fabrizio Catanese, [IBS-KAIST Seminar] Geometry and Dynamics of Geometric Endomorphisms of the Hesse Moduli Space of Elliptic Curves

We consider the geometric map C, called Cayleyan, associating to a plane cubic E the adjoint of its dual curve. We show that C and the classical Hessian map 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 generated by H, C. We point out how the dynamic behaviours of H and C differ drastically. Firstly, concerning the number of real periodic points: for H these are infinitely many, for C they are just 4. Secondly, the Julia set of H is the whole projective line, unlike what happens for all elements of the semigroup which are not iterates of H.