Entropy of automorphisms of II_1-factors arising from the dynamical systems theory

Let a countable amenable group G acts freely and ergodically on a Lebesgue space (X,mu), preserving the measure mu. If T is an automorphism of the equivalence relation defined by G then T can be extended to an automorphism alpha_T of the II_1-factor M=L^\infty(X,\mu)\rtimes G. We prove that if T commutes with the action of G then H(alpha_T)=h(T), where H(alpha_T) is the Connes- Stormer entropy of alpha_T, and h(T) is the Kolmogorov-Sinai entropy of T. We prove also that for given s and t, 0\le s\le t\le\infty, there exists a T such that h(T)=s and H(alpha_T)=t.Comment: LaTeX2e, 12 page

Sub-additive ergodic theorems for countable amenable groups

In this paper we generalize Kingman's sub-additive ergodic theorem to a large class of infinite countable discrete amenable group actions.Comment: Journal of Functional Analysi

T-property and nonisomorphic full factors of types II and III

AbstractThe actions of T-groups on von Neumann's hyperfinite algebras are studied. It is proved that there exist factors of type II with different countable fundamental groups and hence, different actions. It is also proved that there exist nonisomorphic full factors of type III1 with any fixed invariant Sd

Model theory of operator algebras III: Elementary equivalence and II_1 factors

We use continuous model theory to obtain several results concerning isomorphisms and embeddings between II_1 factors and their ultrapowers. Among other things, we show that for any II_1 factor M, there are continuum many nonisomorphic separable II_1 factors that have an ultrapower isomorphic to an ultrapower of M. We also give a poor man's resolution of the Connes Embedding Problem: there exists a separable II_1 factor such that all II_1 factors embed into one of its ultrapowers.Comment: 16 page

Dynamical entropy in Banach spaces

We introduce a version of Voiculescu-Brown approximation entropy for isometric automorphisms of Banach spaces and develop within this framework the connection between dynamics and the local theory of Banach spaces discovered by Glasner and Weiss. Our fundamental result concerning this contractive approximation entropy, or CA entropy, characterizes the occurrence of positive values both geometrically and topologically. This leads to various applications; for example, we obtain a geometric description of the topological Pinsker factor and show that a C*-algebra is type I if and only if every multiplier inner *-automorphism has zero CA entropy. We also examine the behaviour of CA entropy under various product constructions and determine its value in many examples, including isometric automorphisms of l_p spaces and noncommutative tensor product shifts.Comment: 33 pages; unified approach to last three sections give

A theoretical foundation for multi-scale regular vegetation patterns

Self-organized regular vegetation patterns are widespread and thought to mediate ecosystem functions such as productivity and robustness, but the mechanisms underlying their origin and maintenance remain disputed. Particularly controversial are landscapes of overdispersed (evenly spaced) elements, such as North American Mima mounds, Brazilian murundus, South African heuweltjies, and, famously, Namibian fairy circles. Two competing hypotheses are currently debated. On the one hand, models of scale-dependent feedbacks, whereby plants facilitate neighbours while competing with distant individuals, can reproduce various regular patterns identified in satellite imagery. Owing to deep theoretical roots and apparent generality, scale-dependent feedbacks are widely viewed as a unifying and near-universal principle of regular-pattern formation despite scant empirical evidence. On the other hand, many overdispersed vegetation patterns worldwide have been attributed to subterranean ecosystem engineers such as termites, ants, and rodents. Although potentially consistent with territorial competition, this interpretation has been challenged theoretically and empirically and (unlike scale-dependent feedbacks) lacks a unifying dynamical theory, fuelling scepticism about its plausibility and generality. Here we provide a general theoretical foundation for self-organization of social-insect colonies, validated using data from four continents, which demonstrates that intraspecific competition between territorial animals can generate the large-scale hexagonal regularity of these patterns. However, this mechanism is not mutually exclusive with scale-dependent feedbacks. Using Namib Desert fairy circles as a case study, we present field data showing that these landscapes exhibit multi-scale patterning-previously undocumented in this system-that cannot be explained by either mechanism in isolation. These multi-scale patterns and other emergent properties, such as enhanced resistance to and recovery from drought, instead arise from dynamic interactions in our theoretical framework, which couples both mechanisms. The potentially global extent of animal-induced regularity in vegetation-which can modulate other patterning processes in functionally important ways-emphasizes the need to integrate multiple mechanisms of ecological self-organization

Reactive Uptake of Sulfur Dioxide and Ozone on Volcanic Glass and Ash at Ambient Temperature

The atmospheric impacts of volcanic ash from explosive eruptions are rarely considered alongside those of volcanogenic gases/aerosols. While airborne particles provide solid surfaces for chemical reactions with trace gases in the atmosphere, the reactivity of airborne ash has seldom been investigated. Here we determine the total uptake capacity (NiM) and initial uptake coefficient (γM) for sulfur dioxide (SO2) and ozone (O3) on a compositional array of volcanic ash and glass powders at ~25°C in a Knudsen flow reactor. The measured ranges of NiSO2 and γSO2 (1011–1013 molecules cm−2 and 10−3–10−2) and NiO3 and γO3 (1012–1013 molecules cm−2 and 10−3–10−2) are comparable to values reported for mineral dust. Differences in ash and glass reactivity toward SO2 and O3 may relate to varying abundances of, respectively, basic and reducing sites on these materials. The typically lower SO2 and O3 uptake on ash compared to glass likely results from prior exposure of ash surfaces to acidic and oxidizing conditions within the volcanic eruption plume/cloud. While sequential uptake experiments overall suggest that these gases do not compete for reactive surface sites, SO2 uptake forming adsorbed S(IV) species may enhance the capacity for subsequent O3 uptake via redox reaction forming adsorbed S(VI) species. Our findings imply that ash emissions may represent a hitherto neglected sink for atmospheric SO2 and O3