On Endomorphisms of a Solenoid

Geometrically simple Bernoulli generators are constructed for certain ergodic endomorphisms of solenoids. An arbitrary ergodic solenoidal group automorphism is obtained as the limit of a sequence of such Bernoulli factors and hence, by a theorem of D. S. Ornstein, must be measuretheoretically isomorphic to a Bernoulli shift. In his survey paper [6], B. Weiss stated that, using Y. Katznelson's methods, he can prove that every ergodic automorphism of a solenoid is isomorphic to a Bernoulli shift. The aim of this note is to give an alternative proof of this result, with a partial result in the endomorphism case. The methods used are similar to those of L. M. Abramov [1], who used geometrically simple generating partitions in order to compute the entropy of certain solenoidal automorphisms. A comparison will show that Abramov's generators are refinements of the Bernoulli generators exhibited in §2 below. The author would like to acknowledge S. M. Rudolfer, who supervised this work, and R. K. Thomas who suggested the problem to us. For brevity, a working knowledge of measure theory, ergodic theory and topological groups is assumed in what follows, apart from the following fundamental definition of a Bernoulli shift. A measure preserving map ) such that (i) {<b~'P}, i =\ 0, is an independent family of partitions, and (ii) \/j^0<b~'P is the point partition of X. If <b is invertible, (ii) becomes (ii)' V^cb'P is the point partition of X. 1. Details of solenoids are well documented (see e.g. [1], [3] and [2, Chapter VIII). The following brief characterisation will be subsequently useful. Definition 1.1. Let G be a noncyclic subgroup of the discrete additive group Q of rational numbers. The character group 2 of G, called a (onedimensional) solenoid, is a compact, separable, commutative topological group. Proposition 1.2. Let a = iax,a2,...) be a sequence of integers a, ^ 2. Let Ga be the subgroup of Q generated by the elements 11"= l 1/a,-, for n =\ 1. Up to isomorphism, every additive subgroup G of Q, as in 1.1, can be represented as a Ga for some such a. // a is the constant sequence on some integer a, then C7a is the group of a-ary rationals, denoted Ga. Received by the editors December 4, 1974. AMS (MOS) subject classifications (1970).Primary 28A65; Secondary 22D40, 22D45.