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

## September 8 @ 11:00 am - 12:00 pm KST

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*.