Regular polynomial automorphisms in the space of planar quadratic rational maps

Abstract

In this paper, we describe the semistable quotient of the set of regular polynomial automorphisms $${\mathcal {H}}_2^2$$ in the semistable locus of the moduli space of quadratic rational maps, using the portrait moduli space of rational maps with a fixed point. We also provide the parametrization of $${\mathcal {H}}_2^2$$ using two invariants, $$\det f$$ and $${\text {tr}}\,f$$ .