CRITERIA FOR d̄-CONTINUITY

Bernoullicity is the strongest mixing property that a measuretheoretic dynamical system can have. This is known to be intimately connected to the so-called d̄ metric on processes, introduced by Ornstein. In this paper, we consider families of measures arising in a number of contexts and give conditions under which the measures depend d̄-continuously on the parameters. At points where there is d̄-continuity, it is often straightforward to establish that the measures have the Bernoulli property. Introduction One of the most standard examples of a measure-preserving transformation in ergodic theory is that of a Bernoulli shift. To construct these, one starts with a finite set S equipped with a measure m and defines the shift transformation T on the space X = SZ equipped with the measure mZ, the direct product of copies of m indexed by the integers. These transformations model the simple situation which occurs in probability theory of having a sequence of independent identically distributed random variables (Bernoulli trials). For this reason, Bernoulli shifts are often referred to as the most random possible dynamical systems. A fundamental question which often arises in ergodic theory is when a pair of measure-preserving transformations are measure-theoretically isomorphic. These questions are known to be hard, and until the mid 1970s, the results were few. It then came as a great surprise when Ornstein showed in a series of papers (collected together as [11] see also [16]) that there is a very simple classification of Bernoulli shifts (namely, that two Bernoulli shifts are isomorphic if and only if they have the same entropy) and gave verifiable conditions for checking whether other systems are also isomorphic to Bernoulli shifts. This resulted in an explosion of work, showing that a large number of ‘naturally occurring’ measure-preserving systems were isomorphic to Bernoulli shifts. When this is the case, it is often said that the system has the Bernoulli property. Some of the results in this paper fall into this category. A convenient way of establishing Bernoullicity in our context is provided by means of the d̄ distance introduced by Ornstein. A useful description of this is offered by joinings (see Rudolph’s book [14] for details). A joining of (X, ν1) and (X, ν2) is a shift-invariant measure on X ×X with the property that μ(A×X) = ν1(A) and μ(X × B) = ν2(B) for any measurable subsets A and B of X . Writing Received by the editors March 7, 1996 and, in revised form, September 18, 1996. 1991 Mathematics Subject Classification. Primary 28D05, 60G10.