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New normal forms for degree-3 polynomials and rational functions

Heidi Benham, Alexander Galarraga, Benjamin Hutz, Joey Lupo, Wayne Peng and Adam Towsley

Vol. 16 (2023), No. 4, 605–620

When studying families in the moduli space of dynamical systems, choosing an appropriate representative function for a conjugacy class can be a delicate task. The most delicate questions surround rationality of the conjugacy class compared to rationality of the defining polynomials of the representation. We give a normal form for degree-3 polynomials which has the property that the set of fixed points is equal to the set of fixed point multipliers. This normal form is given in terms of moduli space invariants and, hence, has nice rationality properties. We further classify all degree-3 rational maps which can be conjugated to have a similar relationship between the fixed points and the fixed point multipliers.

dynamical system, normal form, moduli space
Mathematical Subject Classification
Primary: 37P05, 37P45
Secondary: 37P15
Received: 28 April 2022
Revised: 20 October 2022
Accepted: 27 October 2022
Published: 31 October 2023

Communicated by Amanda Folsom
Heidi Benham
Department of Mathematics
Western Oregon University
Monmouth, OR
United States
Alexander Galarraga
Department of Mathematics
University of Washington
Seattle, WA
United States
Benjamin Hutz
Department of Mathematics and Statistics
Saint Louis University
St.Louis, MO
United States
Joey Lupo
Department of Mathematics and Statistics
Amherst College
Amherst, MA
United States
Wayne Peng
Department of Mathematics
University of Rochester
Rochester, NY
United States
Adam Towsley
School of Mathematical Sciences
Rochester Institute of Technology
Rochester, NY
United States