Showing papers in "Bulletin of The London Mathematical Society in 1977"

Exceptional Lie Algebras and Related Algebraic and Geometric Structures

Abstract: Certain algebraic structures, most notably associative, alternative, and Jordan algebras are strongly linked via construction and classification to simple Lie algebras and to interesting geometries. These geometries are in turn linked to simple Lie algebras via their groups of collineations. These linkages serve to illustrate how various notions of exceptionality in algebra and geometry (e.g., non-classical Lie algebras, non-associative alternative algebras, non-special Jordan algebras, and nonDesarguian projective planes) are just different manifestations of the same phenomenon. It is the intent of this survey to discuss briefly the general classes of structures in which the exceptional objects occur, to describe the linkage between the exceptional objects, and to illustrate the utility of these linkages in understanding the nature of these diverse exceptional structures.

A Finitary Classification of Topological Markov Chains and Sofic Systems

Abstract: In [9] a classification theory for topological Markov chains (shifts of finite type, intrinsic Markov chains) appears which, although incomplete [10], provides algebraic criteria for deciding when two such chains are topologically conjugate. Numerous invariants effectively exclude topological conjugacy for many interesting examples. However, as mentioned in [9], the main algebraic test (shift equivalence of matrices) is probably noncomputable in general. For this reason it seems appropriate to find an alternative weaker classification. The main purpose of this paper is to define \" finite equivalence \" and to show that two (irreducible) topological Markov chains are so related if and only if they have the same topological entropy. This result generalizes immediately, in view of a theorem of Coven and Paul [5] to the sofic systems of Weiss [8]. Finite equivalence involves surjective continuous maps which are finite to one (in fact a.e. n to 1 for some n and bounded to one). In view of the relationship between Axiom A diffeomorphisms and topological Markov chains c.f. [1], [3], [7], there are good reasons for considering equivalences which involve non-invertible continuous maps. We have in mind especially, the fact that every Axiom A diffeomorphism restricted to its non-wandering set is a continuous (a.e. one-one) image of a topological Markov chain. An immediate consequence of this fact and our main theorem is that two such restricted diffeomorphisms with the same topological entropy are finite to one images of a common topological Markov chain. We conclude the paper by showing that any measure preserving continuous map which semi-conjugates two irreducible finite state Markov chains (not necessarily with maximal measures) is finite to one if and only if these chains have the same entropy. This is a generalization of a theorem of Coven and Paul [4]. My thanks are due to R. Adler, E. Coven, H. Furstenberg and W. Goodwyn, visitors to Warwick this summer, for helpful discussions concerning various classification problems related to this paper. Furstenberg's Lemma 2 is crucial to the proof of our main theorem.