No abstract

We consider Ledrappier's dynamical system, which was the first example of a Z 2 -action which is 2-mixing but not 3-mixing. Our main result is that, excluding certain small "constructible" sets, the system is mixing of every order.

A representation field for a non-maximal order H in a central simple algebra is a subfield of the spinor class field of maximal orders which determines the set of spinor genera of maximal orders containing a copy of H. Not every non-maximal order has a representation field. In this work we prove that every commutative order has a representation field and give a formula for it. The main result is proved for central simple algebras over arbitrary global fields. Corps de representations pour ordres commutatifsRésumé. Un corps de représentation pour un ordre non maximal H dans une algèbre centrale simple est un sous-corps du corps de classes espinoriel d'ordres maximaux qui détermine l'ensemble de genres espinoriels d'ordres maximaux qui contiennent un conjugué de H. Un ordre non maximal ne possède pas forcement un corps de représentation. Dans ce travail nous montrons que chaque ordre commutatif a un corps de représentation F et nous donnons une formule pour F . Le résultat principal est prouvé pour des algèbres simples centrales sur des corps globaux arbitraires.

If H ⊆ D are two orders in a central simple algebra A with D of maximal rank, the representation field F (D|H) is a subfield of the spinor class field of the genus of D which determines the set of spinor genera of orders in that genus representing the order H. Previous work have focused on two cases, maximal orders D and commutative orders H. In this work, we show how to compute the representation field F (D|H) when D is the intersection of a finite family of maximal orders, e.g., an Eichler order, and H is arbitrary. Examples are provided.with the following properties:

We apply the theory of Bruhat-Tits trees to the study of optimal embeddings of two and three dimensional commutative orders into quaternion algebras. Specifically, we determine how many conjugacy classes of global Eichler orders in a quaternion algebra yield optimal representations of such orders. This completes the previous work by C. Maclachlan, who considered only Eichler orders of square free level and integral domains as sub-orders. The same technique is used in the second part of this work to compute local embedding numbers, extending previous results by J. Brzezinski.