Cocycles and the structure of ergodic group actions

We show, for a large class of groups, the existence of cocycles taking values in these groups and which define ergodic skew products. We apply this to prove a generalization of Ambrose’s representation theorem for ergodic actions of these groups.     

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.     

Skew-product extensions of Markov operators and products of dependent random variables

In this paper we generalize the notion of skew products as known in ergodic theory to skew product extensions of Markov operators. We prove that Markov operators are of such a type iff they have relative discrete spectrum (in a slightly generalized sense) thus generalizing a theorem of Parry. In addition we show that skew product extensions of Markov operators play an important role in the theory of products of dependent random variables and we develop this interdependence between the two theories thus generalizing results of Koutsky, Schmetterer and Wolff.     

An example of compact extensions of Kronecker factors and skew products of irrational rotations with finite groups

We give an example to show that compact extensions of Kronecker factors for two ergodic commuting measure preserving transformations can be different. Also, a criteria for ergodicity of skew products of irrational rotations with finite abelian groups is obtained.     

The structure of skew products with ergodic group automorphisms

We prove that ergodic automorphisms of compact groups are Bernoulli shifts, and that skew products with such automorphisms are isomorphic to direct products. We give a simple geometric demonstration of Yuzvinskii’s basic result in the calculation of entropy for group automorphisms, and show that the set of possible values for entropy is one of two alternatives, depending on the answer to an open problem in algebraic number theory. We also classify those algebraic factors of a group automorphism that are complemented.     

Attracting graphs of skew products with non-contracting fiber maps

We study attracting graphs of step skew products from the topological and ergodic points of view where the usual contracting-like assumptions of the fiber dynamics are replaced by weaker merely topological conditions. In this context, we prove the existence of an attracting invariant graph and study its topological properties. We prove the existence of globally attracting measures and we show that (in some specific cases) the rate of convergence to these measures is exponential.

     

Skew products over rotations with exotic properties

We show that a $\text{ Z}_{2}$ skew product of a badly approximable rotation can be minimal and not uniquely ergodic. This construction is used to construct a Z skew product of a rotation where the orbit of a.e. point is dense but Lebesgue measure is not ergodic.     

绕积马氏链的几个结果

本文利用一般马氏链的理论讨论了随机环境中的马氏链的各种状态的特征及遍历性,并用两种方式将状态空间进行严格的分类.     

Persistence of nonhyperbolic measures for <Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript>-diffeomorphisms

In the space of diffeomorphisms of an arbitrary closed manifold of dimension ≥ 3, we construct an open set such that each difteomorphism in this set has an invariant ergodic measure with respect to which one of its Lyapunov exponents is zero. These difteomorphisins are constructed to have a partially hyperbolic invariant set on which the dynamics is conjugate to a soft skew product with the circle as the fiber. It is the central Lyapunov exponent that proves to be zero in this case, and the construction is based on an analysis of properties of the corresponding skew products.     

Onμ-recurrent nonsingular endomorphisms

We show that the Maharam skew product ofμ-recurrent nonsingular endomorphisms is conservative and give some applications. Among them is the construction of a conservative ergodic invertible natural extension forμ-recurrent ergodic nonsingular endomorphisms. Supported in part by a Williams College Faculty Research Grant.     

Normal ergodic actions and extensions

We demonstrate that normal ergodic extensions of group actions are characterized as skew product actions given by cocycles into locally compact groups. As a consequence, Robert Zimmer’s characterization of normal ergodic group actions is generalized to the noninvariant case. We also obtain the uniqueness theorem which generalizes the von Neumann Halmos uniqueness theorem and Zimmer’s uniqueness theorem for normal actions with relative discrete spectrum.     

A SKEW-PRODUCT WITH SIMPLE SINGULAR SPECTRUM

Abstract

Using the method of approximation in Ergodic Theory we first prove a general result giving conditions for a skew product to be ergodic and have simple and singular spectrum. Using this result we explicitly construct a skew product over a rotation with rotations in the tibres which is ergodic and has simple and singular spectrum.     

On Dynamics Related to a Class of Numeration Systems

 In this paper, we investigate the class of numeration systems and we study the associated dynamical systems, called odometers. It is shown that these odometers are measure-theoretically isomorphic to rank one transformations on the unit interval, constructed by a cutting-stacking method. Furthermore, a symbolic coding leads to isomorphic shift systems arising from substitutions. Some skew products of the odometers by cocycles related to the sum of digits are shown to be ergodic.     

Skew Products and Ergodic Theorems for Group Actions

New ergodic theorems for the action of a free semigroup on a probabilistic space by measure-preserving maps are obtained. The method applied consists of associating with the original semigroup action a skew product over the shift on the space of infinite one-sided sequences of generators of the semigroup and then integrating the BirkhoffKhinchin ergodic theorems along the base of the skew product. Bibliography: 17 titles.     

On Dynamics Related to a Class of Numeration Systems

 In this paper, we investigate the class of numeration systems and we study the associated dynamical systems, called odometers. It is shown that these odometers are measure-theoretically isomorphic to rank one transformations on the unit interval, constructed by a cutting-stacking method. Furthermore, a symbolic coding leads to isomorphic shift systems arising from substitutions. Some skew products of the odometers by cocycles related to the sum of digits are shown to be ergodic. Received 5 March 2001; in revised form 16 August 2001     

Split skew products,a related functional equation,and specification

We prove that a skew product of a measure-preserving transformation with an ergodic automorphism of a compact abelian group is always isomorphic to their direct product via an isomorphism that merely translates the group fibers. This requires solving a functional equation. A weak version of Bowen’s specification property is essential to our construction of a solution.     

Hamiltonian loops from the ergodic point of view

Let G be the group of Hamiltonian diffeomorphisms of a closed symplectic manifold Y. A loop h:S¹→G is called strictly ergodic if for some irrational number α the associated skew product map T:S¹×Y→S¹×Y defined by T(t,y)=(t+α,h(t)y) is strictly ergodic. In the present paper we address the following question. Which elements of the fundamental group of G can be represented by strictly ergodic loops? We prove existence of contractible strictly ergodic loops for a wide class of symplectic manifolds (for instance for simply connected ones). Further, we find a restriction on the homotopy classes of smooth strictly ergodic loops in the framework of Hofer’s bi-invariant geometry on G. Namely, we prove that their asymptotic Hofer’s norm must vanish. This result provides a link between ergodic theory and symplectic topology. Received July 7, 1998 / final version received September 14, 1998     

Invariant cocycles, random tilings and the super- and strong Markov properties

We consider -cocycles with values in locally compact, second countable abelian groups on discrete, nonsingular, ergodic equivalence relations. If such a cocycle is invariant under certain automorphisms of these relations, we show that the skew product extension defined by the cocycle is ergodic. As an application we obtain an extension of many recent results of the author and K. Petersen to higher-dimensional shifts of finite type, and prove a transitivity result concerning rearrangements of certain random tilings.

     

Ergodic properties of skew products in infinite measure

Let (Ω, µ) be a shift of finite type with a Markov probability, and (Y, ν) a non-atomic standard measure space. For each symbol i of the symbolic space, let Φ_i be a non-singular automorphism of (Y, ν). We study skew products of the form (ω, y) ? (σω, Φ_ω0 (y)), where σ is the shift map on (Ω, µ). We prove that, when the skew product is recurrent, it is ergodic if and only if the Φ_i’s have no common non-trivial invariant set.     

Absence of C 1- Ω-Explosion in the Space of Smooth Simplest Skew Products

We give a detailed proof of absence of a C ¹- Ω-explosion in the space of C ¹-regular simplest skew products of mappings of an interval (i.e., skew products of mappings of an interval with a closed set of periodic points). We study the influence of C ¹-perturbations (of the class of skew products) to the set of periods of the periodic points of C ¹-regular simplest skew products, and describe the peculiarities of period doubling bifurcations of the periodic points.