scholarly journals Weak containment of measure-preserving group actions

2019 ◽  
Vol 40 (10) ◽  
pp. 2681-2733 ◽  
Author(s):  
PETER J. BURTON ◽  
ALEXANDER S. KECHRIS
Keyword(s):  
Global Structure ◽  
Equivalence Classes ◽  
The Other ◽  
Unitary Representations ◽  
Weak Equivalence ◽  
Interesting Fact ◽  
Weak Containment ◽  
Probability Spaces ◽  
Measure Preserving Actions ◽  
Standard Probability

This paper concerns the study of the global structure of measure-preserving actions of countable groups on standard probability spaces. Weak containment is a hierarchical notion of complexity of such actions, motivated by an analogous concept in the theory of unitary representations. This concept gives rise to an associated notion of equivalence of actions, called weak equivalence, which is much coarser than the notion of isomorphism (conjugacy). It is well understood now that, in general, isomorphism is a very complex notion, a fact which manifests itself, for example, in the lack of any reasonable structure in the space of actions modulo isomorphism. On the other hand, the space of weak equivalence classes is quite well behaved. Another interesting fact that relates to the study of weak containment is that many important parameters associated with actions, such as the type, cost, and combinatorial parameters, turn out to be invariants of weak equivalence and in fact exhibit desirable monotonicity properties with respect to the pre-order of weak containment, a fact that can be useful in certain applications. There has been quite a lot of activity in this area in the last few years, and our goal in this paper is to provide a survey of this work.

Approximate equivalence of group actions

2017 ◽  
Vol 38 (4) ◽  
pp. 1201-1237 ◽  
Author(s):  
ANDREAS NÆS AASERUD ◽  
SORIN POPA
Keyword(s):  
Group Actions ◽  
The Other ◽  
Equivalence Relations ◽  
Orbit Equivalence ◽  
Usual Type ◽  
New Concepts ◽  
Probability Spaces ◽  
Measure Preserving Actions

We consider several weaker versions of the notion of conjugacy and orbit equivalence of measure preserving actions of countable groups on probability spaces, involving equivalence of the ultrapower actions and asymptotic intertwining conditions. We compare them with the other existing equivalence relations between group actions, and study the usual type of rigidity questions around these new concepts (superrigidity, calculation of invariants, etc).

scholarly journals The Howe-Moore property for real and $p$-adic groups

2011 ◽  
Vol 109 (2) ◽  
pp. 201 ◽  
Author(s):  
Raf Cluckers ◽  
Yves Cornulier ◽  
Nicolas Louvet ◽  
Romain Tessera ◽  
Alain Valette
Keyword(s):  
Lie Groups ◽  
Ergodic Measure ◽  
Oscillatory Integrals ◽  
The Other ◽  
Unitary Representations ◽  
Compact Groups ◽  
Locally Compact ◽  
Relative Version ◽  
Closed Normal Subgroup ◽  
Measure Preserving Actions

We consider in this paper a relative version of the Howe-Moore property, about vanishing at infinity of coefficients of unitary representations. We characterize this property in terms of ergodic measure-preserving actions. We also characterize, for linear Lie groups or $p$-adic Lie groups, the pairs with the relative Howe-Moore property with respect to a closed, normal subgroup. This involves, in one direction, structural results on locally compact groups all of whose proper closed characteristic subgroups are compact, and, in the other direction, some results about the vanishing at infinity of oscillatory integrals.

Weak Containment and Induced Representations of Groups

1962 ◽  
Vol 14 ◽  
pp. 237-268 ◽  
Author(s):  
J. M. G. Fell
Keyword(s):  
Dual Space ◽  
Locally Compact Group ◽  
Equivalence Classes ◽  
Unitary Representations ◽  
Unitary Equivalence ◽  
Locally Compact ◽  
Induced Representations ◽  
Weak Containment ◽  
Irreducible Unitary Representations ◽  
Special Case

Let G be a locally compact group and G† its dual space, that is, the set of all unitary equivalence classes of irreducible unitary representations of G. An important tool for investigating the group algebra of G is the so-called hull-kernel topology of G†, which is discussed in (3) as a special case of the relation of weak containment. The question arises: Given a group G, how do we determine G† and its topology? For many groups G, Mackey's theory of induced representations permits us to catalogue all the elements of G†. One suspects that by suitably supplementing this theory it should be possible to obtain the topology of G† at the same time. It is the purpose of this paper to explore this possibility. Unfortunately, we are not able to complete the programme at present.

On the cohomology equivalence classes for non-singular mappings

1998 ◽  
Vol 18 (6) ◽  
pp. 1385-1397
Author(s):  
ISAAC KORNFELD ◽  
ANDREI KRYGIN
Keyword(s):  
Ergodic Theorem ◽  
Equivalence Classes ◽  
Probability Spaces

The structure of the cohomology equivalence classes for non-singular, not necessarily invertible mappings of probability spaces is studied. In particular, some results of Kochergin and Ornstein–Smorodinsky on the structure of these classes for measure-preserving automorphisms are generalized to the case of non-singular endomorphisms. Our approach is based on Hopf's maximal ergodic theorem and its proof by Garsia.

Regent Results in Comma-Free Codes

1963 ◽  
Vol 15 ◽  
pp. 178-187 ◽  
Author(s):  
B. H. Jiggs
Keyword(s):  
Equivalence Class ◽  
Cyclic Permutation ◽  
Equivalence Classes ◽  
The Other ◽  
Complete Class

A set D of k-letter words is called a comma-free dictionary (2), if whenever (a1a2 . . . ak) and (b1b2 . . . bk) are in D, the "overlaps" (a2a3 . . . akb1), (a3a4 . . . akb1b2), . . . , (akb1 . . . bk-1) are not in D. We say that two k-letter words are in the same equivalence class if one is a cyclic permutation of the other. An equivalence class is called complete if it contains k distinct members. Comma-freedom is violated if we choose words from incomplete equivalence classes, or if more than one word is chosen from the same complete class.

scholarly journals Measure-theoretic chaos

2012 ◽  
Vol 34 (1) ◽  
pp. 110-131 ◽  
Author(s):  
TOMASZ DOWNAROWICZ ◽  
YVES LACROIX
Keyword(s):  
Ergodic Measure ◽  
Uniform Measure ◽  
Ergodic System ◽  
Topological System ◽  
Ergodic Measures ◽  
Topological Chaos ◽  
Probability Spaces ◽  
Entropy Zero ◽  
Standard Probability

AbstractWe define new isomorphism invariants for ergodic measure-preserving systems on standard probability spaces, called measure-theoretic chaos and measure-theoretic$^+$ chaos. These notions are analogs of the topological chaos DC2 and its slightly stronger version (which we denote by $\text {DC}1\frac 12$). We prove that: (1) if a topological system is measure-theoretically (measure-theoretically$^+$) chaotic with respect to at least one of its ergodic measures then it is topologically DC2 $(\text {DC}1\frac 12)$ chaotic; (2) every ergodic system with positive Kolmogorov–Sinai entropy is measure-theoretically$^+$ chaotic (even in a slightly stronger uniform sense). We provide an example showing that the latter statement cannot be reversed, that is, of a system of entropy zero with uniform measure-theoretic$^+$chaos.

scholarly journals Equivalence of Semantics in Argumentation

2021 ◽  
Author(s):  
Leila Amgoud ◽  
Vivien Beuselinck
Keyword(s):  
Fibonacci Numbers ◽  
Evaluation Methods ◽  
The Other ◽  
Weak Equivalence ◽  
Exponential Series ◽  
Grounded Semantics

A large number of evaluation methods, called semantics, have been proposed in the literature for assessing strength of arguments. This paper investigates their equivalence. It argues that for being equivalent, two semantics should have compatible evaluations of both individual arguments and pairs of arguments. The first requirement ensures that the two semantics judge an argument in the same way, while the second states that they provide the same ranking of arguments. We show that the two requirements are completely independent. The paper introduces three novel relations between semantics based on their rankings of arguments: weak equivalence, strong equivalence and refinement. They state respectively that two semantics do not disagree on their strict rankings; the rankings of the semantics coincide; one semantics agrees with the strict comparisons of the second and it may break some of its ties. We investigate the properties of the three relations and their links with existing principles of semantics, and study the nature of relations between most of the existing semantics. The results show that the main extensions semantics are pairwise weakly equivalent. The gradual semantics we considered are pairwise incompatible, however some pairs are strongly equivalent in case of flat graphs including Max-based (Mbs) and Euler-based (Ebs), for which we provide full characterizations in terms respectively of Fibonacci numbers and the numbers of an exponential series. Furthermore, we show that both semantics (Mbs, EMbs) refine the grounded semantics, and are weakly equivalent with the other extension semantics. We show also that in case of flat graphs, the two gradual semantics Trust-based and Iterative Schema characterize the grounded semantics, making thus bridges between gradual semantics and extension semantics. Finally, the other gradual semantics are incompatible with extension semantics.

scholarly journals Nano topology induced by Lattices

2019 ◽  
Vol 38 (5) ◽  
pp. 197-204
Author(s):  
M. Lellis Thivagar ◽  
V. Sutha Devi
Keyword(s):  
Lower Bound ◽  
19Th Century ◽  
Upper Bound ◽  
Lattice Theory ◽  
Ordered Set ◽  
Equivalence Classes ◽  
The Other ◽  
Five Elements ◽  
Lower And Upper Approximations ◽  
The 19Th Century

Lattice is a partially ordered set in which all finite subsets have a least upper bound and greatest lower bound. Dedekind worked on lattice theory in the 19th century. Nano topology explored by Lellis Thivagar et.al. can be described as a collection of nano approximations, a non-empty finite universe and empty set for which equivalence classes are buliding blocks. This is named as Nano topology, because of its size and what ever may be the size of universe it has atmost five elements in it. The elements of Nano topology are called the Nano open sets. This paper is to study the nano topology within the context of lattices. In lattice, there is a special class of joincongruence relation which is defined with respect to an ideal. We have defined the nano approximations of a set with respect to an ideal of a lattice. Also some properties of the approximations of a set in a lattice with respect to ideals are studied. On the other hand, the lower and upper approximations have also been studied within the context various algebraic structures.

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