New York Journal of Mathematics
Volume 27 (2021), 1613-1702


Brandon Gontmacher, Benjamin Hutz, Grayson Jorgenson, Srinjoy Srimani, and Simon Xu

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

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Published: December 10, 2021.
Keywords: dynamical system, automorphism group, moduli space, preperiodic points.
Subject: 37P45, 37P05, 37P15.

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.

Author information

Brandon Gontmacher:
Department of Mathematics
University of Maryland
College Park, MD 20742, USA


Benjamin Hutz:
Department of Mathematics and Statistics
Saint Louis University
St. Louis, MO 63103, USA


Grayson Jorgenson:
Department of Mathematics
Florida State University
Tallahassee, FL 32306, USA


Srinjoy Srimani:
Department of Mathematics
Brown University
Providence, RI 02912, USA


Simon Xu:
Department of Mathematics and Statistics
Colby College
Waterville, Maine 04901, USA