Automorphism loci for degree 3 and degree 4 endomorphisms of the projective line

Abstract

Let $f$ be an endomorphism of the projective line. There is a natural conjugation action on the space of such morphisms by elements of the projective linear group. The group of automorphisms, or stabilizer group, of a given $f$ for this action is known to be a finite group. We determine explicit families that parameterize all endomorphisms defined over $\bar{\mathbb{Q}}$ of degree $3$ and degree $4$ that have a nontrivial automorphism, the \textit{automorphism locus} of the moduli space of dynamical systems. We analyze the geometry of these loci in the appropriate moduli space of dynamical systems. Further, for each family of maps, we study the possible structures of $\mathbb{Q}$-rational preperiodic points which occur under specialization.

Many corrections made. Code for computations included in this version. 132 pages

References in corpus (2)

• The Moduli Space of Cubic Rational Maps
• Periodic points and tail lengths of split polynomial maps modulo primes

Cited by in corpus (1)

• Automorphism Groups of Endomorphisms of ${\mathbb{P}}^1({\bar{\mathbb{F}}}_p)$