Ergodic affine transformations are loosely Bernoulli

We prove that an ergodic affine transformation of a compact abelian group is loosely Bernoulli, that is, it can be induced from a Bernoulli shift.     

A loosely Bernoulli counterexample machine

In Rudolph’s paper on minimal self joinings [7] he proves that a rank one mixing transformation constructed by Ornstein [5] can be used as the building block for many ergodic theoretical counterexamples. In this paper we show that Ornstein’s transformation can be altered to create a general method for producing zero entropy, loosely Bernoulli counter-examples. This paper answers a question posed by Ornstein, Rudolph, and Weiss [6].     

A non-Bernoulli skew product which is loosely Bernoulli

LetT: Y→Y be the Bernoulli two shift with independent generatorQ={Q ₀,Q ₁} and letS: X→X be a measure preserving bijection. If (S, X) is ergodic then the skew product onX×Y defined by {fx339-1} is aK-automorphism. IfŜ is also Bernoulli we sayS is pre-Bernoulli. J. Feldman showed that ifS is pre-Bernoulli thenS must be loosely Bernoulli. We construct an example to show the converse is false, i.e. anS that is loosely Bernoulli but not pre-Bernoulli.     

Finitely fixed implies loosely Bernoulli,a direct proof

As defined in the literature, a process is loosely Bernoulli if a certain propertyP(ɛ) is satisfied for every ɛ>0. Using only facts about stationary joinings of processes, it is shown that given ɛ>0 there exists δ>0 such that whenever two processes are separated by less than δ in the -metric and one of them is loosely Bernoulli, the other is “almost” loosely Bernoulli in the sense thatP(ɛ) is satisfied. As easy corollaries, one has that loosely Bernoulli processes are closed in the and that finitely fixed processes are loosely Bernoulli. Research supported by NSF Grant MCS-78-21335 and the Joint Services Electronics Program under Contract N00014-79-C-0424.     

The cartesian square of the horocycle flow is not loosely Bernoulli

We give an example of an algebraic non-loosely Bernoulli flow. Namely, we prove that the cartesian square of the classical horocycle flow is not loosely Bernoulli. Partially supported by the Sloan Foundation and NSF Grant MCS74-19388.     

A zero-entropy mixing transformation whose product with itself is loosely Bernoulli

A zero-entropy mixing transformationT is constructed such thatT×T is loosely Bernoulli (LB). Previously known examples were not mixing. The construction is then generalized to yield a zero-entropy mixing transformationT such that then-fold productT × … ×T is LB for each positive integern. Furthermore, a flow with the same properties is obtained.     

The relative isomorphism theorem for Bernoulli flows

In this paper we extend the work of Thouvenot and others on Bernoulli splitting of ergodic transformations to ergodic flows of finite entropy. We prove that ifA is a factor of a flowS, whereS ₁ is ergodic andA has a Bernoulli complement inS ₁, thenA has a Bernoulli complement inS. Consequently, Bernoulli splitting for flows is stable under taking intermediate factors and certain limits. In addition it follows that the property of isomorphism with a Bernoulli × zero entropy flow is similarly stable.     

Ergodic automorphisms ofT n are Bernoulli shifts

An automorphism of then-dimensional torusT ⁿ, none of whose eigenvalues is a root of unity includes on the canonical measure space ofT ⁿ a measure preserving transformation which is isomorphic to a Bernoulli shift. This research was supported in part by the National Science Foundation grant GP-18884 and by the European Research Office of the U.S. Army contract DAJA-37-70-C-0701.     

Smooth partitions of Anosov diffeomorphisms are weak Bernoulli

It is shown that smooth partitions are weak Bernoulli forC ² measure preserving Anosov diffeomorphisms. A related type of coding is defined and an invariant discussed. Supported by the Sloan Foundation and NSF GP-14519.     

Geodesic flows are Bernoullian

A geometric method is developed for proving that transformations are isomorphic to Bernoulli shifts. The method is applied to the geodesic flows on surfaces of negative curvature and it is shown that they are isomorphic to Bernoulli flows. This result was announced in [9] but the argument envisioned then was incomplete.     

Two nonisomorphic flows with the same very weak Bernoulli partition on a base

We build two flows with the same very weak Bernoulli partition on a base that are not isomorphic even after time scaling. Both flows are built under functions which take constant values on each of the atoms of the partiton. Moreover the values of each function are independent over the rationals.     

Identities for Bernoulli polynomials and Bernoulli numbers

We prove that if m and \({\nu}\) are integers with \({0 \leq \nu \leq m}\) and x is a real number, then

1. $$\sum_{k=0 \atop k+m \, \, odd}^{m-1} {m \choose k}{k+m \choose \nu} B_{k+m-\nu}(x) = \frac{1}{2} \sum_{j=0}^m (-1)^{j+m} {m \choose j}{j+m-1 \choose \nu} (j+m) x^{j+m-\nu-1},$$ where B _n (x) denotes the Bernoulli polynomial of degree n. An application of (1) leads to new identities for Bernoulli numbers B _n . Among others, we obtain
2. $$\sum_{k=0 \atop k+m \, \, odd}^{m -1} {m \choose k}{k+m \choose \nu} {k+m-\nu \choose j}B_{k+m-\nu-j} =0 \quad{(0 \leq j \leq m-2-\nu)}. $$ This formula extends two results obtained by Kaneko and Chen-Sun, who proved (2) for the special cases j = 1, \({\nu=0}\) and j = 3, \({\nu=0}\) , respectively.