Entropy Theory from the Orbital Point of View

 Inspired by [17], we develop an orbital approach to the entropy theory for actions of countable amenable groups. This is applied to extend – with new short proofs – the recent results about uniform mixing of actions with completely positive entropy [17], Pinsker factors and the relative disjointness problems [10], Abramov–Rokhlin entropy addition formula [19], etc. Unlike the cited papers our work is independent of the standard machinery developed by Ornstein–Weiss [14] or Kieffer [12]. We do not use non-orbital tools like the Rokhlin lemma, the Shannon–McMillan theorem, castle analysis, joining techniques for amenable actions, etc. which play an essential role in [17], [19] and [10]. (Received 23 October 2000)     

Sofic mean dimension

We introduce mean dimensions for continuous actions of countable sofic groups on compact metrizable spaces. These generalize the Gromov–Lindenstrauss–Weiss mean dimensions for actions of countable amenable groups, and are useful for distinguishing continuous actions of countable sofic groups with infinite entropy.     

Local entropy theory for a countable discrete amenable group action

The local properties of entropy for a countable discrete amenable group action are studied. For such an action, a local variational principle for a given finite open cover is established, from which the variational relation between the topological and measure-theoretic entropy tuples is deduced. While doing this it is shown that two kinds of measure-theoretic entropy for finite Borel covers coincide. Moreover, two special classes of such an action: systems with uniformly positive entropy and completely positive entropy are investigated.     

An entropy-preserving Dye’s theorem for ergodic actions

This work shows that equality of entropy for ergodic actions of a discrete amenable group is a restricted orbit equivalence in the formal sense defined inRestricted Orbit Equivalence for Actions of Discrete Amenable Groups by Kammeyer and Rudolph [3]. An element of the full-group of such an action encodes a countable partition. A natural extension of entropy to such countable partitions is shown to be a size in the sense of [3] and hence engenders and orbit equivalence relation on the space of all such actions. The major goal achieved here is to show that this relation is precisely equality of entropy.     

Local variational principle concerning entropy of a sofic group action

Recently Lewis Bowen introduced a notion of entropy for measure-preserving actions of countable sofic groups admitting a generating measurable partition with finite entropy; and then David Kerr and Hanfeng Li developed an operator-algebraic approach to actions of countable sofic groups not only on a standard probability space but also on a compact metric space, and established the global variational principle concerning measure-theoretic and topological entropy in this sofic context. By localizing these two kinds of entropy, in this paper we prove a local version of the global variational principle for any finite open cover of the space, and show that these local measure-theoretic and topological entropies coincide with their classical counterparts when the acting group is an infinite amenable group.     

可数sofic群的等距线性作用的维数

我们对复Banach空间上的可数sofic群的等距线性作用提出了一种新的维数，推广了复Banach空间上的可数顺从群的等距线性作用的Voiculescu维数，并且在可数sofic群的情形回答了Gromov的一个问题.     

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.     

The Rohlin tower theorem and hyper-finiteness for actions of continuous groups

We prove the Rohlin tower theorem for free measure preserving actions of locally compact second countable solvable groups and almost connected amenable groups. This theorem was known for l.c.s.c. abelian groups and was recently extended by Ornstein and Weiss to discrete solvable groups. We extend their methods to the continuous case, using the structure theory of the class of groups under consideration. As a corollary we obtain that free actions of such groups generate hyperfinite equivalence relations. Work supported in part by NSF grant MCS 74-19876. A02.     

Entropy Theory for Cross-Sections

We define the notion of entropy for a cross-section of an action of continuous amenable group, and relate it to the entropy of the ambient action. As a result, we are able to answer a question of J.P. Thouvenot about completely positive entropy actions.     

Generators for amenable group actions

A generalisation of Krieger's finite generator theorem is proved for free actions of countable amenable groups on a non-atomic Lebesgue probability space.     

Polynomials with a common composite

We develop a conditional entropy theory for infinite measure preserving actions of countable discrete amenable groups with respect to a σ-finite factor. This includes ‘infinite’ analogues of relative Kolmogorov-Sinai, Rokhlin and Krieger theorems on generating partitions, Pinsker theorem on disjointness, Furstenberg decomposition and disjointness theorems, etc. In case of ℤ-action, our concept of relative entropy matches well the ‘absolute’ entropy h _Kr introduced by Krengel. Answering in part his question and a question of Silva and Thieullen, we show that for any non-distal transformation S there exists an infinite measure preserving transformation T with h _Kr(T × S) = ∞ but h _Kr(T) = 0. This project was supported in part by a CRDF grant UM1-2546-KH-03.     

Entropy and the variational principle for actions of sofic groups

Recently Lewis Bowen introduced a notion of entropy for measure-preserving actions of a countable sofic group on a standard probability space admitting a generating partition with finite entropy. By applying an operator algebra perspective we develop a more general approach to sofic entropy which produces both measure and topological dynamical invariants, and we establish the variational principle in this context. In the case of residually finite groups we use the variational principle to compute the topological entropy of principal algebraic actions whose defining group ring element is invertible in the full group C ^∗-algebra.     

Some permanence properties of C-unique groups

We will study some permanence properties of C^*-unique groups in details. In particular, normal subgroups and extensions will be considered. Among other interesting results, we prove that every second countable amenable group with an injective finite-dimensional representation (not necessarily unitary) is a retract of a C^*-unique group. Moreover, any amenable discrete group is a retract of a discrete C^*-unique group.     

Numerical Characteristics of Groups and Corresponding Relations

The entropy of the random walk on a discrete countable group can be used for comparison of systems of generators. The fundamental inequality between growth, entropy, and escape rate gives possibility of defining ``the best' system of generators. We formulate a new circle of problems related to growth and other asymptotical characteristics of groups. Bibliography: 13 titles.     

The Abramov-Rokhlin entropy addition formula for amenable group actions

In this note we show that the entropy of a skew product action of a countable amenable group satisfies the classical formula of Abramov and Rokhlin.     

On Twisted Fourier Analysis and Convergence of Fourier Series on Discrete Groups

We study norm convergence and summability of Fourier series in the setting of reduced twisted group C ^*-algebras of discrete groups. For amenable groups, Følner nets give the key to Fejér summation. We show that Abel-Poisson summation holds for a large class of groups, including e.g. all Coxeter groups and all Gromov hyperbolic groups. As a tool in our presentation, we introduce notions of polynomial and subexponential H-growth for countable groups w.r.t. proper scale functions, usually chosen as length functions. These coincide with the classical notions of growth in the case of amenable groups.     

From acyclic groups to the Bass conjecture for amenable groups

We prove that the Bost Conjecture on the ¹-assembly map for countable discrete groups implies the Bass Conjecture. It follows that all amenable groups satisfy the Bass Conjecture.     

Approximately transitive dynamical systems and simple spectrum

For some countable discrete torsion Abelian groups we give examples of their finite measure-preserving actions which have simple spectrum and no approximate transitivity property.     

Finite entropy actions of free groups,rigidity of stabilizers,and a Howe-Moore type phenomenon

We study a notion of entropy, called f-invariant entropy, introduced by Lewis Bowen for probability measure preserving actions of finitely generated free groups. In the degenerate case, the f-invariant entropy is -∞. In this paper, we investigate the qualitative consequences of an action having finite f-invariant entropy. We find three main properties of such actions. First, the stabilizers occurring in factors of such actions are highly restricted. Specifically, the stabilizer of almost every point must be either trivial or of finite index. Second, such actions are very chaotic in the sense that when the space is not essentially countable, every non-identity group element acts with infinite Kolmogorov-Sinai entropy. Finally, we show that such actions display behavior reminiscent of the Howe-Moore property. Specifically, if the action is ergodic, there exists an integer n such that for every non-trivial normal subgroup K, the number of K-ergodic components is at most n. Our results are based on a new formula for f-invariant entropy.     

Topological pressure for sub-additive potentials of amenable group actions

We introduce the topological pressure for any sub-additive potentials of a countable discrete amenable group action and any given open cover, and establish a local variational principle for it.