Periodic codings of Bratteli-Vershik systems

We develop conditions for the coding of a Bratteli-Vershik system according to initial path segments to be periodic, equivalently for a constructive symbolic recursive scheme corresponding to a cutting and stacking process to produce a periodic sequence. This is a step toward understanding when a Bratteli-Vershik system can be essentially faithfully represented by means of a natural coding as a subshift on a finite alphabet.

Non-Bernoulli systems with completely positive entropy

A new approach to actions of countable amenable groups with completely positive entropy (cpe), allowing one to answer some basic questions in this field, was recently developed. The question of the existence of cpe actions which are not Bernoulli was raised. In this paper, we prove that every countable amenable groupG, which contains an element of infinite order, has non-Bernoulli cpe actions. In fact we can produce, for any$h \in (0, \infty ]$, an uncountable family of cpe actions of entropyh, which are pairwise automorphically non-isomorphic. These actions are given by a construction which we call co-induction. This construction is related to, but different from the standard induced action. We study the entropic properties of co-induction, proving that ifαGis co-induced from an actionαΓof a subgroup Γ, thenh(αG)=h(αΓ). We also prove that ifαΓis a non-Bernoulli cpe action of Γ, thenαGis also non-Bernoulli and cpe. Hence the problem of finding an uncountable family of pairwise non-isomorphic cpe actions of the same entropy is reduced to one of finding an uncountable family of non-Bernoulli cpe actions of$\mathbb Z$, which pairwise satisfy a property we call ‘uniform somewhat disjointness’. We construct such a family using refinements of the classical cutting and stacking methods.

Numerical Mixing Analysis of a Vaned Circular Micromixer

Mixing of fluids is a common and often critical step in microfluidic systems. In typical large scale processes turbulence greatly speeds the mixing process. At the mini and micro-scales, however, the flow is laminar and the benefits of turbulent mixing are not present. Mixing at the mini- and micro-scales tends to become a more highly engineered process of bringing fluids together in predictable ways to achieve a predetermined and acceptable level of mixing. This paper summarizes a numerical analysis of the mixing performance of a vaned circular micromixer. A newly developed mixing metric suitable for reacting fluids is developed for this study. Applying the basic steps of stretching, cutting, and stacking to effect mixing, a useful micromixer is analyzed numerically for its mixing efficiency. A parametric study of flow and viscosity indicate that a flow Re of 12 or higher is sufficient to achieve effective and rapid mixing in this device.

A map with topological minimal self-joinings in the sense of del Junco

Andrés del Junco has proposed a definition of topological minimal self-joinings intended to parallel Dan Rudolph's measure-theoretic concept. By means of a rank-two ‘cutting and stacking’, this article constructs the first example of a system (a subshift) satisfying his proposed definition of 2-fold topological minimal self-joinings.The second part of the article shows that 2-fold topological minimal self-joinings does not imply 3-fold and that no map has 4-fold topological minimal self-joinings. This latter result follows from a generalization of a theorem of Schwartzman.