Multipliers and invariants of endomorphisms of projective space in dimension greater than 1

Hutz, Benjamin

Journal de théorie des nombres de Bordeaux, Tome 32 (2020) no. 2, pp. 439-469.

Il existe une action par conjugaison naturelle sur l’ensemble des endomorphismes de ${\mathbb{P}}^N$ de degré fixé $d\ge 2.$ L’ensemble quotient pour cette action forme l’espace de modules des endomorphismes de degré $d$ de ${\mathbb{P}}^N$, que l’on note ${ℳ}_d^N$. Nous construisons des fonctions invariantes sur ces espaces de modules, qui proviennent de l’ensemble des matrices des multiplicateurs des points périodiques. Nous démontrons des propriétés élémentaires de ces fonctions, en particulier qu’elles sont régulières sur ${ℳ}_d^N$, et établissons des méthodes pour les calculer, ainsi que l’existence de relations entre elles. Dans la partie principale de l’article, on étudie dans quelle mesure ces fonctions invariantes déterminent la classe de conjugaison dans l’espace de modules. Des types différents des familles isospectrales sont construits, et une généralisation du théorème de McMullen sur l’application multiplicateur de dimension $1$ est proposée. Finalement, nous prouvons que le dernier résultat est vrai aussi pour certaines familles spécifiques dans ${ℳ}_d^N$.

There is a natural conjugation action on the set of endomorphism of ${\mathbb{P}}^N$ of fixed degree $d\ge 2$. The quotient by this action forms the moduli of degree $d$ endomorphisms of ${\mathbb{P}}^N$, denoted ${ℳ}_d^N$. We construct invariant functions on this moduli space coming from the set of multiplier matrices of the periodic points. The basic properties of these functions are demonstrated such as that they are in the ring of regular functions of ${ℳ}_d^N$, methods of computing them, as well as the existence of relations. The main part of the article examines to what extent these invariant functions determine the conjugacy class in the moduli space. Several different types of isospectral families are constructed and a generalization of McMullen’s theorem on the multiplier mapping of dimension 1 is proposed. Finally, this generalization is shown to hold when restricted to several specific families in ${ℳ}_d^N$.

Classification : 37P45, 37P05, 37A35
Mots clés : dynamical systems, multiplier invariants, moduli space

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