We study the discrete dynamics of standard (or left) polynomials $f(x)$ over division rings D. We define their fixed points to be the points $\lambda \in D$ for which $f^{\circ n}(\lambda )=\lambda $ for any $n \in \mathbb {N}$ , where $f^{\circ n}(x)$ is defined recursively by $f^{\circ n}(x)=f(f^{\circ (n-1)}(x))$ and $f^{\circ 1}(x)=f(x)$ . Periodic points are similarly defined. We prove that $\lambda $ is a fixed point of $f(x)$ if and only if $f(\lambda )=\lambda $ , which enables the use of known results from the theory of polynomial equations, to conclude that any polynomial of degree $m \geq 2$ has at most m conjugacy classes of fixed points. We also show that in general, periodic points do not behave as in the commutative case. We provide a sufficient condition for periodic points to behave as expected.

We provide a complete classification of possible graphs of rational preperiodic points of endomorphisms of the projective line of degree 2 defined over the rationals with a rational periodic critical point of period 2, under the assumption that these maps have no periodic points of period at least 7. We explain how this extends results of Poonen on quadratic polynomials. We show that there are exactly 13 possible graphs, and that such maps have at most nine rational preperiodic points. We provide data related to the analogous classification of graphs of endomorphisms of degree 2 with a rational periodic critical point of period 3 or 4.

Let φ be a an endomorphism of degree d ≥ 2 of the projective line, defined over a number field K. Let S be a finite set of places of K, including the archimedean places, such that φ has good reduction outside of S. The article presents two main results: the first result is a bound on the number of K-rational preperiodic points of φ in terms of the cardinality of the set S and the degree d of the endomorphism φ. This bound is quadratic in terms of d which is a significant improvement to all previous bounds on the number of preperiodic points in terms of the degree d. For the second result, if we assume that there is a K-rational periodic point of period at least two, then there exists a bound on the number of K-rational preperiodic points of φ that is linear in terms of the degree d.

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L'accès aux articles de la revue « Journal de Théorie des Nombres de Bordeaux » (http://jtnb.cedram.org/), implique l'accord avec les conditions générales d'utilisation (http://jtnb.cedram. org/legal/). Toute reproduction en tout ou partie de cet article sous quelque forme que ce soit pour tout usage autre que l'utilisation à fin strictement personnelle du copiste est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Journal de Théorie des Nombres de Bordeaux 31 (2019), 49-79 Quadratic rational functions with a rational periodic critical point of period 3 par Solomon VISHKAUTSAN Avec un appendice par Michael Stoll Résumé. Nous établissons une classification complète des graphes des points rationnels prépériodiques des fonctions rationnelles de degré 2 ayant un point critique rationnel de période 3 sous les hypothèses suivantes : ces fonctions n'admettent pas de points de période supérieure à 5 et une certaine conjecture sur le nombre de points rationnels sur une courbe affine plane de genre 6 est vraie. Nous montrons qu'il y a exactement six graphes possibles et que les fonctions associées possèdent au plus onze points prépériodiques.