Geometric Endomorphisms of the Hesse moduli space of elliptic curves

TL;DR

This paper explores the geometric and dynamic properties of two maps, Cayleyan and Hessian, on elliptic curves, revealing their algebraic relations, fixed points, and differing behaviors in dynamics and real periodic points.

Contribution

It introduces the semigroup generated by Cayleyan and Hessian maps, analyzing their geometric and dynamic interactions on elliptic curves, and identifies special fixed elliptic curves within this framework.

Findings

The maps generate a free semigroup.

Elliptic curves fixed by endomorphisms of the semigroup are identified.

Different dynamic behaviors of the maps are demonstrated, especially in real periodic points and Julia sets.

Abstract

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 $\mathcal W(\frak H, \frak C)$ generated by $ \frak H, \frak 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 $\mathcal W (\frak H, \frak C)$ which are not iterates of $ \mathfrak H$.