Showing papers in "Proceedings of The London Mathematical Society in 1984"

Wandering Domains in the Iteration of Entire Functions

Abstract: Fonctions entieres a domaines multiplement connexes de normalite. Ensembles de Fatou-Julia pour des fonctions qui commuttent. Exemples de type d'Herman. Classes de fonctions entieres sans domaines d'errance

Congruence Properties of Zariski‐Dense Subgroups I

Abstract: This paper deals with the following general situation: we are given an algebraic group G defined over a number field K, and a subgroup F of the group G(K) of K-rational points of G. Then what should it mean for F to be a 'large' subgroup? We might require F to be a lattice in G, to be arithmetic, to contain many elements of a specific kind, to have a large closure in some natural topology, et cetera. There are many theorems proving implications between conditions of'size' of this kind. We shall consider the case of G a K-simple group, usually Q-simple. The Zariski topology of G, for which the closed sets are those defined by the vanishing of polynomial functions, is very coarse. On the other hand, the various valuations v of K each give rise to a 'strong' topology on G(K); if v is non-archimedean, and Kv is the completion of K with respect to v, then G(KV) is a non-archimedean Lie group, and if £>„ is the ring of integers of Kv, the open subgroup G(£5 J of G(KV) of integral points is defined for all but finitely many v. The Zariski closure F of a subgroup F of G(K) is a /C-algebraic subgroup of G, while the strong closure Fy is a Lie subgroup of G(KV). For elementary reasons the dimension of Tv is at most the dimension of F. Our results are in the opposite direction: we show, under suitable conditions on G, that if F = G then for almost all v we have that Tv contains G(OV). To illustrate this with a specific example, if Ml 5. . . , Mk are matrices in SL2(Q) generating a group F which is Zariskidense in SL2, then for all sufficiently large prime numbers p we have F p = SL2(Zp). Further, this result is effective, in the sense that we could, in principle, check the hypothesis for given Mx,..., Mk, and derive a bound on p beyond which the conclusion holds. (It is perhaps worth remarking that this special case, which is the simplest application of our results, may be proved with much less effort than the general theorem.)

Binary Convexities and Distributive Lattices

Abstract: A convex structure is binary if every finite family of pairwise intersecting convex sets has a non-empty intersection. Distributive lattices with the convexity of all order-convex sublattices are a prominent type of example, because they correspond exactly to the intervals of a binary convex structure which has a certain separation property. In one direction, this result relies on a study of so-called base-point orders induced by a convex structure. These orderings are used to construct an 'intrinsic' topology. For binary convexities, certain basic questions are answered with the aid of some results on completely distributive lattices. Several applications are given. Dimension problems are studied in a subsequent paper. 0. Introduction A convex structure consists of a set X, together with a collection ^ of subsets of X, henceforth called convex sets, such that (1.1) the empty set and the set X are convex; (1.2) the intersection of convex sets is convex; (1.3) the union of an updirected collection of convex sets is convex. The collection € itself is called a convexity on X. Axiom (1.2) allows the construction of an associated (convex) hull operator (usually denoted by h) in the obvious way. The hull of a finite set is called a polytope, and the hull of a two-point set is also called an interval. A half-space is a convex set with a convex complement. The following separation axioms—comparable with the axioms Tl 5 . . . ,T4, in topology—are used frequently: Sx: singletons are convex (which we will assume throughout); S2: two distinct points are in complementary half-spaces; S3: a convex set is an intersection of half-spaces; S4: two disjoint convex sets extend to complementary half-spaces. In this paper we will concentrate largely on a particular, though fundamental, class of convexities with the following binarity property: each finite collection of pairwise intersecting convex sets has a non-empty intersection. The basic types of examples are described in §1. Many of these examples arise in a topological context; for other examples, a natural topology can be constructed. Binary convex structures on a topological space have been studied extensively in the past years [15, 17, 19, 22, 29, 34, 38]. The topology and the convexity are always assumed to be compatible in the sense that polytopes are closed. A triple, consisting of a set X, a convexity €, and a compatible topology ST, is called a topological convex Proc. London Math. Soc. (3), 48 (1984), 1-33. 5388.3.48 A

Dilation of a Dissipative Quantum Dynamical System to a Quantum Markov Process

Abstract: On considere une C-etoile algebre #7B-A qui est une algebre de Von Neumann #7B-M, et P t est un semigroupe d'applications completement positives normales preservant l'identite avec un etat normal invariant. On obtient un processus de Markov non commutatif au sens de Accardi avec une dilatation du semigroupe vers un groupe d'automorphisme implante unitaire

Dimension of Binary Convex Structures

Abstract: It is shown that for compact spaces with a normal binary convexity, the dimension functions dim, ind, Ind, and cohomological dimension are all equal. Also, an H-dimensional compact space X with a normal binary convexity embeds in a product of an M-dimensional connected quotient of X with the space of Xcomponents. Both results are obtained with the aid of an auxiliary theory of convex dimension. Some applications to completely distributive lattices are given. 0. Introduction For a connected and completely distributive lattice it is known [2, 3.4] that the Lebesgue covering dimension 'dim' and the cohomological dimension cdG (where G is an abelian coefficient group) are equal. By [3, Corollary 1], the small inductive dimension 'ind' equals cdG provided that, in addition, the set of points at which the lattice attains its small inductive dimension has non-void interior. We have taken up the equality problem for the various dimension functions at the following general level. Let X be a normal binary (that is, with Helly number at most 2) convex structure with compact polytopes. According to [32], all intervals of X are completely distributive lattices under suitably chosen base-point orders. On the other hand, the space X with its weak topology embeds as a median-stable subset of a Tychonov cube. Our first main result is as-follows. 0.1. THEOREM. / / X is a compact space with a normal binary convexity, then for each coefficient group G, cdGX = dim X = ind X = Ind X. If, instead of being compact, X is only required to have compact polytopes, and if A" is Lindelof in its weak topology, then it is still true that dimX = indX. In [34], a result similar to Theorem 0.1 has been obtained for a restricted class of Lawson semilattices. Also, there is a theorem of Pasynkov [23, p. 722 of the English translation] asserting that dim = ind = Ind for locally compact groups. One is tempted to conclude that dimension functions tend to coincide on topological algebras (see [32, 3.6] for the algebraicity of binary convexities). The equality theorem, 0.1, actually involves one other dimension function: the socalled convex dimension 'cind'. This function, defined on topological convex structures only, was studied in [28] for convex structures with connected convex sets, and it will be the key tool in proving Theorem 0.1. For convexities with connected convex sets, but with an unrestricted Helly number, an equality theorem of the above type has already been obtained in [22], where cohomological dimension was not considered. To eliminate the connectedness restriction, we have re-examined 'cind' in the case of non-connected convex sets, and it appears that the main results of [28] survive for Proc. London Math. Soc. (3), 48 (1984), 34-54. DIMENSION OF BINARY CONVEX STRUCTURES 35 binary convex structures. This almost self-contained treatment of'cind' is given in § 1. A proof of Theorem 0.1 is given in §2, together with some consequences. For instance, we use results of [29] concerning convexity invariants to re-prove the well-known equality of breadth and dimension for connected completely distributive lattices (see [2, Corollary 3.4; 3, Corollary 1; 17, Corollary 2.4]). Our second main result is the following. 0.2. THEOREM. Let X be an n-dimensional compact space with a normal binary convexity. Then there exists a convexity-preserving quotient q: X -* qX onto an n-dimensional connected space qX with a normal binary convexity, such that q, together with the decomposition map d: X -> dX (the space of components), gives an embedding of topological convex structures