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Determination of all rational preperiodic points for morphisms of PN


Author: Benjamin Hutz
Journal: Math. Comp. 84 (2015), 289-308
MSC (2010): Primary 37P05, 37P15; Secondary 37P45, 37-04
DOI: https://doi.org/10.1090/S0025-5718-2014-02850-0
Published electronically: May 5, 2014
MathSciNet review: 3266961
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Abstract: For a morphism $ f:\mathbb{P}^N \to \mathbb{P}^N$, the points whose forward orbit by $ f$ is finite are called preperiodic points for $ f$. This article presents an algorithm to effectively determine all the rational preperiodic points for $ f$ defined over a given number field $ K$. This algorithm is implemented in the open-source software Sage for $ \mathbb{Q}$. Additionally, the notion of a dynatomic zero-cycle is generalized to preperiodic points. Along with examining their basic properties, these generalized dynatomic cycles are shown to be effective.


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Additional Information

Benjamin Hutz
Affiliation: Department of Mathematical Sciences, Florida Institute of Technology, 150 W. University Boulevard, Melbourne, Florida 32901
Email: bhutz@fit.edu

DOI: https://doi.org/10.1090/S0025-5718-2014-02850-0
Keywords: Dynamical systems, rational preperiodic points, uniform boundedness, Poonen's conjecture, algorithm
Received by editor(s): November 8, 2012
Received by editor(s) in revised form: April 18, 2013
Published electronically: May 5, 2014
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.