Infinite partitions and Rokhlin towers

Abstract

We find a countable partition P on a Lebesgue space, labeled {1, 2, 3, . . .}, for any non-periodic measure-preserving transformation T such that P generates T and, for the T, P process, if you see an n on time −1 then you only have to look at times −n, 1 − n, . . . −1 to know the positive integer i to put at time 0. We alter that proof to extend every non-periodic T to a uniform martingale (i.e. continuous g function) on an infinite alphabet. If T has positive entropy and the weak Pinsker property, this extension can be made to be an isomorphism. We pose remaining questions on uniform martingales. In the process of proving the uniform martingale result we make a complete analysis of Rokhlin towers which is of interest in and of itself. We also give an example that looks something like an independent identically distributed process on Z 2 when you read from right to left but where each column determines the next if you read left to right.Please see the dedication to Dan Rudolph that we included in the paper, 'Non-intersecting splitting σ -algebras in a non-Bernoulli transformation', also in this edition.Countable and uncountable partitions can be strange. For processes generated by a finite alphabet, one definition of zero entropy is that a process has zero entropy if and only if the past determines the future. Parry [11, Theorem 8.2, p. 90] showed that every transformation has a countable generator such that the past determines the future. (For the present paper, generator means that the whole process (past and future) generates but Parry uses the word generator to mean that the past generates.) Here we improve on that result with Theorem 1.1 below.Definition 1. Throughout this paper T is a one-to-one measure-preserving transformation on a Lebesgue space with probability measure P such that P({ω : T i (ω) = ω for any i}) = 0. This is what we mean by a non-periodic transformation.Comment 2. If T is one-to-one and measure-preserving on a Lebesgue space then T −1 so is (measurability of T −1 is standard theory but not obvious) and if T is non-periodic then so is T −1 .Definition 2. When T is a measure-preserving transformation and P is a partition, the P, T process is the process in which every ω ∈ is assigned to a doubly infinite word . . . X −2 , X −1 , X 0 , X 1 , X 2 , . . . where X i is the element of P containing T i (ω). The word it maps ω to is called its T, P name. The T, P process is the process of T, P names endowed with the measure induced by the map from points to T, P names. Since T is measure preserving it is a stationary process.Comment 3. Let T be a measure-preserving transformation P a partition and . . . X −2 , X −1 , X 0 , X 1 , X 2 , . . . be the P, T process, then if the P, T name of ω is . . . a −2 , a −1 , a 0 , a 1 , a 2 , . . . then the P, T name of T (ω) is . . . b −2 , b −1 , b 0 , b 1 , b 2 , . . . defined by b i := a i+1 for all i, so the map from ω to its doubly infinite name is a homomorphism onto the doubly infinite sequences of elements of P with induced measure where the transfor...

Tower multiplexing and slow weak mixing

“…In [22], results are presented on rates of approximation by periodic transformations, and connections with dynamical properties. Recent research of Kalikow demonstrates the utility of developing a general theory of Rokhlin towers [20]. Also, it is clear from the Kakutani-Rokhlin lemma that any ergodic measure preserving tranformation can be approximated arbitrarily well by another ergodic measure preserving transformation from any isomorphism class.…”

Independence and Alpern multitowers

“…The result described in (1) was used most famously in the proof by Ornstein of his Isomorphism Theorem for Bernoulli shifts ( [Orn70]), and has proven useful to others over time (see [Kor04] for a broader discussion). The Alpern tower result has also been extensively applied ([EP97] has a brief overview, see [Kal12] for a more recent application), and has been shown to be an equivalent form of two other important results in ergodic theory ([AP08]).…”