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 Q of degree 3 and degree 4 that have a nontrivial automorphism, the 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 Q-rational preperiodic points which occur under specialization.

The authors thank the Institute for Computational and Experimental Research in Mathematics which oversaw the REU during the summer of 2019, where the majority of this project was completed. Also the Brown Center for Computation and Visualization which provided computational resources for some of the computations. Several other individuals provide helpful discussions and expertise on various specific rational points problems in this paper: Brendan Hassett, Michael Stoll, Andrew Sutherland, Jan Tuitman, Jennifer Balakrishnan, and an anonymous referee.