Bowen entropy for actions of amenable groups

Bowen introduced a definition of topological entropy of subset inspired by Hausdorff dimension in 1973 [1]. In this paper we consider the Bowen entropy for amenable group action dynamical systems and show that, under the tempered condition, the Bowen entropy of the whole compact space for a given Følner sequence equals the topological entropy. For the proof of this result, we establish a variational principle related to the Bowen entropy and the Brin–Katok local entropy formula for dynamical systems with amenable group actions.     

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.     

Entropy increase for ℤd-actions on a Lebesgue space

Our aim is to introduce the concepts of the entropy increase and the asymptotic rate of entropy increase for a ?^d-action on a Lebesgue space and to show that for ergodic ?^d-actions the asymptotic rate of entropy increase coincides with the Conze—Katznelson—Weiss (CKW) entropy. The result is the multidimensional analogue of the Goldstein result for onedimensional dynamical systems.     

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.     

A counterexample of the entropy of a skew product

We have a well known formula for the entropy of a skew product in the case where the fiber action is generated from a Z-action. Although not much is known about the skew product with a more general group, we are able to compute the entropy of a skew product whose fiber action comes from a special type of Z²-actions [Pa1]: The entropy of the skew product is known to be the entropy of the base plus the directional entropy in the direction of the integral of the skewing function. We present an example of a skew product to show that the above computation of the entropy is not true for a general Z²-action on the fiber.     

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.     

Typical ℤn-actions can be inserted only in injective ℝn-actions

We study actions of the groups ?ⁿ and ?ⁿ by Lebesgue space automorphisms. We prove that a typical ?ⁿ-action can be inserted only in injective actions of ?ⁿ, n ∈ ?. We give a simple proof of the fact that a typical ?²-action cannot be inserted in an ?-action.     

Amenable ergodic group actions and an application to Poisson boundaries of random walks

We introduce and study the class of amenable ergodic group actions which occupy a position in ergodic theory parallel to that of amenable groups in group theory. We apply this notion to questions about skew products, the range (i.e., Poincaré flow) of a cocycle, and to Poisson boundaries.     

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.     

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.     

Ellis semigroupoid

We extend the notion of the enveloping semigroup of a locally compact group to the enveloping semigroupoid of a locally compact groupopid and show that there is a universal enveloping semigroupoid which is unique up to isomorphism. As in the group case, we associate the Ellis semigroupoid to an action of a locally compact groupoid on a fibrewise compact space. We define the notion of proximality for groupoid actions and characterize it in terms of Ellis semigroupoid.

     

Entropy of geometric structures

We give a notion of entropy for general gemetric structures, which generalizes well-known notions of topological entropy of vector fields and geometric entropy of foliations, and which can also be applied to singular objects, e.g. singular foliations, singular distributions, and Poisson structures. We show some basic properties for this entropy, including the additivity property, analogous to the additivity of Clausius-Boltzmann entropy in physics. In the case of Poisson structures, entropy is a new invariant of dynamical nature, which is related to the transverse structure of the characteristic foliation by symplectic leaves.     

On sofic actions and equivalence relations

The notion of sofic equivalence relation was introduced by Gabor Elek and Gabor Lippner. Their technics employ some graph theory. Here we define this notion in a more operator algebraic context, starting from Connes? Embedding Problem, and prove the equivalence of these two definitions. We introduce a notion of sofic action for an arbitrary group and prove that an amalgamated product of sofic actions over amenable groups is again sofic. We also prove that an amalgamated product of sofic groups over an amenable subgroup is again sofic.     

The Adjoint Action of an Expansive Algebraic -Action

 Let , and let α be an expansive -action by continuous automorphisms of a compact abelian group X with completely positive entropy. Then the group of homoclinic points of α is countable and dense in X, and the restriction of α to the α-invariant subgroup is a -action by automorphisms of . By duality, there exists a -action by automorphisms of the compact abelian group : this action is called the adjoint action of α. We prove that is again expansive and has completely positive entropy, and that α and are weakly algebraically equivalent, i.e. algebraic factors of each other. A -action α by automorphisms of a compact abelian group X is reflexive if the -action on the compact abelian group adjoint to is algebraically conjugate to α. We give an example of a non-reflexive expansive -action α with completely positive entropy, but prove that the third adjoint is always algebraically conjugate to . Furthermore, every expansive and ergodic -action α is reflexive. The last section contains a brief discussion of adjoints of certain expansive algebraic -actions with zero entropy. Received 11 June 2001; in revised form 29 November 2001     

Mean dimension, small entropy factors and an embedding theorem

In this paper we show how the notion of mean dimension is connected in a natural way to the following two questions: what points in a dynamical system (X, T) can be distinguished by factors with arbitrarily small topological entropy, and when can a system (X, T) be embedded in (([0, 1] ^d ) ^Z , shift). Our results apply to extensions of minimalZ-actions, and for this case we also show that there is a very satisfying dimension theory for mean dimension.     

On purely non-free finite actions of abelian groups on compact surfaces

A finite group of self-homeomorphisms of a closed orientable surface is said to act on it purely non-freely if each of its elements has a fixed point; we also call it a gpnf-action. In this paper we observe that gpnf-actions exist for an arbitrary finite group and we discuss the minimum genus problem for such actions. We solve it for abelian groups. In the cyclic case we prove that the minimal gpnf-action genus coincides with Harvey’s minimal genus.     

Attractors of reaction‐diffusion systems in unbounded domains and their spatial complexity

The nonlinear reaction‐diffusion system in an unbounded domain is studied. It is proven that, under some natural assumptions on the nonlinear term and on the diffusion matrix, this system possesses a global attractor ?? in the corresponding phase space. Since the dimension of the attractor happens to be infinite, we study its Kolmogorov's ?‐entropy. Upper and lower bounds of this entropy are obtained. Moreover, we give a more detailed study of the attractor for the spatially homogeneous RDE in ?ⁿ. In this case, a group of spatial shifts acts on the attractor. In order to study the spatial complexity of the attractor, we interpret this group as a dynamical system (with multidimensional “time” if n > 1) acting on a phase space ??. It is proven that the dynamical system thus obtained is chaotic and has infinite topological entropy. In order to clarify the nature of this chaos, we suggest a new model dynamical system that generalizes the symbolic dynamics to the case of the infinite entropy and construct the homeomorphic (and even Lipschitz‐continuous) embedding of this system into the spatial shifts on the attractor. Finally, we consider also the temporal evolution of the spatially chaotic structures in the attractor and prove that the spatial chaos is preserved under this evolution. © 2003 Wiley Periodicals, Inc.     

On the mean square of the remainder for the euclidean lattice point counting problem

We study an invariant of dynamical systems called naive entropy, which is defined for both measurable and topological actions of any countable group. We focus on nonamenable groups, in which case the invariant is two-valued, with every system having naive entropy either zero or infinity. Bowen has conjectured that when the acting group is sofic, zero naive entropy implies sofic entropy at most zero for both types of systems. We prove the topological version of this conjecture by showing that for every action of a sofic group by homeomorphisms of a compact metric space, zero naive entropy implies sofic entropy at most zero. This result and the simple definition of naive entropy allow us to show that the generic action of a free group on the Cantor set has sofic entropy at most zero. We observe that a distal Γ-system has zero naive entropy in both senses, if Γ has an element of infinite order. We also show that the naive entropy of a topological system is greater than or equal to the naive measure entropy of the same system with respect to any invariant measure.     

The Spectrum of Completely Positive Entropy Actions of Countable Amenable Groups

We prove that an ergodic free action of a countable discrete amenable group with completely positive entropy has a countable Lebesgue spectrum. Our approach is based on the Rudolph-Weiss result on the equality of conditional entropies for actions of countable amenable groups with the same orbits. Relative completely positive entropy actions are also considered. An application to the entropic properties of Gaussian actions of countable discrete abelian groups is given.     

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)