Ergodic decompositions associated with regular Markov operators on Polish spaces

Few, if any, of the properties enjoyed by ergodic measure-preserving transformations defined on a finite measure space generalize in a natural way to those defined on an infinite measure space. Concrete examples of ergodic transformations which preserve a finite measure and ones which preserve an infinite measure exist in the literature, see [1]. It is not difficult to see that ergodic transformations never admit wandering sets of positive measure. In [2 ] it was shown that a basic difference exists between ergodic transformations which preserve a finite measure and those which preserve an infinite measure; namely, an ergodic measure-preserving transformation defined on an infinite measure space always admits weakly wandering sets of positive measure (Theorem 2 of [2]). Unlike wandering sets, it is not true in general that the union of two weakly wandering sets is again a weakly wandering set even if we require that a class of mutually disjoint images of one weakly wandering set does not intersect a class of mutually disjoint images of the other. One may ask then if there are any ergodic measure-preserving transformations defined on an infinite measure space which admit only weakly wandering sets of finite measure. In this paper we show that this is not the case. We prove that there always exist weakly wandering sets of infinite measure for any ergodic measure-preserving transformation defined on an infinite measure space (Theorem 3). In [2] the existence of a weakly wandering set of positive measure for an ergodic measure-preserving transformation defined on an infinite measure space was discovered while studying the necessary and sufficient conditions for the existence of a finite, invariant, and equivalent measure for a given measurable and nonsingular transformation. In this paper we construct the weakly wandering sets in a different way. Using the pointwise ergodic theorem we prove a simple and yet a useful fact about ergodic measure-preserving transformations defined on an infinite measure space (Theorem 2). It states that given two sets A and B both of finite measure, it is possible to find an image of A under some power of the transformation T which has small intersection with the set B. This fact is basic in proving Lemma 1 which shows the existence of a

We consider the topology and dynamics associated with a wide class of matchbox manifolds, including spaces of aperiodic tilings and suspensions of higher rank (potentially nonabelian) group actions on zero-dimensional spaces. For such a space we introduce a topological invariant, the homology core, built using an expansion of it as an inverse sequence of simplicial complexes. The invariant takes the form of a monoid equipped with a representation, which in many cases can be used to obtain a finer classification than is possible with the previously developed invariants. When the space is obtained by suspending a topologically transitive action of the fundamental group $ \Gamma $ of a closed orientable manifold on a zero-dimensional compact space $ Z$, this invariant corresponds to the space of finite Borel measures on $ Z$ which are invariant under the action of $ \Gamma $. This leads to connections between the rank of the core and the number of invariant, ergodic Borel probability measures for such actions. We illustrate with several examples how these invariants can be calculated and used for topological classification and how it leads to an understanding of the invariant measures.

The aim of this paper is to prove ergodic decomposition theorems for probability measures which are quasi-invariant under Borel actions of inductively compact groups as well as for -finite invariant measures. For infinite measures the ergodic decomposition is not unique, but the measure class of the decomposing measure on the space of projective measures is uniquely defined by the initial invariant measure.Bibliography: 21 titles.

0. We consider transition probability operators P(x,E) on a compact metric space which satisfy a rather strong condition (uniform stability in mean). ?1 is devoted largely to notation and terminology. In ?2 an ergodic decomposition is obtained. It is essentially a sharpening of a decomposition of Yosida's valid under less stringent conditions [17]. In ?3 we discuss the behavior of sample paths of the Markov processes themselves relative to the ergodic decomposition obtained in ?2. In ?4 we give some examples of transition probability operators which are uniformly stable in mean. Most of the results of ?2 and some of the results of ?4 appeared in the author's thesis, written under the guidance of Professor Blackwell. Tbe results of ?3 had to await Breiman's discovery of a new version of the strong law of largenumbers. Our Theorem 3.2 is a minor generalization of Breiman's theorem and our proof a slight modification of his. 1. S is a compact metric space. Its metric is denoted by d. , is the a-field generated by the open subsets of S. C(S) is the Banach space of real-valued continuous functions on S with the uniform norm. The dual C*(S) consists of all the finite countably additive measures on S. The value of a member p of C*(S) at a member f of C(S) will be denoted by either (f, ,u) or f f dy. A countably additive measure M on any a-field over any set Q is called a probability measure if it is non-negative and if ,u(Q) = 1. We sometimes refer to probability measures in C*(S) as distributions. A function P on S x I is called a transition probability operator if (i) P(x, * ) is a probability distribution for each x E S, (ii) P( , E) is a s-measurable function for each E e Z. Unless otherwise stated, we will from now on assume that P is a fixed transition probability operator. We will abuse the language by using 'P(x,E)' rather than 'P' to denote not only a value of P but P itself. Associated with P(x,E) are operators T and U, the first defined on bounded X-measurable functions f by (Tf)(x)

We prove two theorems in the ergodic theory of infinite permutation groups. First, generalizing a theorem of Nessonov for the infinite symmetric group, we show that every non-singular action of a non-archimedean, Roelcke precompact, Polish group on a measure space $(\Omega , \mu )$ admits an invariant $\sigma $-finite measure equivalent to $\mu $. Second, we prove the following de Finetti-type theorem: if $G \curvearrowright M$ is a primitive permutation group with no algebraicity verifying an additional uniformity assumption, which is automatically satisfied if $G$ is Roelcke precompact, then any $G$-invariant, ergodic probability measure on $Z^{M}$, where $Z$ is a Polish space, is a product measure.

Neretin in [Izv. Ross. Akad. Nauk Ser. Mat. 77 (2013), pp. 95–108] constructed an analogue of the Hua measures on the infinite p p -adic matrices M a t ( N , Q p ) \mathrm {Mat}\left (\mathbb {N},\mathbb {Q}_p\right ) . Bufetov and Qiu in [Compos. Math. 153 (2017), pp. 2482–2533] classified the ergodic measures on M a t ( N , Q p ) \mathrm {Mat}\left (\mathbb {N},\mathbb {Q}_p\right ) that are invariant under the natural action of G L ( ∞ , Z p ) × G L ( ∞ , Z p ) \mathrm {GL}(\infty ,\mathbb {Z}_p)\times \mathrm {GL}(\infty ,\mathbb {Z}_p) . In this paper we solve the problem of ergodic decomposition for the p p -adic Hua measures introduced by Neretin. We prove that the probability measure governing the ergodic decomposition has an explicit expression which identifies it with a Hall-Littlewood measure on partitions. Our arguments involve certain Markov chains.

Stationarity, invariance, and ergodicity; discrete- and continuous-time ergodic theorems; moment and maximum inequalities; multivariate ergodic theorems; sample intensity of a random measure; subadditivity and products of random matrices; conditioning and ergodic decomposition; shift coupling and the invariant σ-field

We study ergodic decompositions of Dirichlet spaces under intertwining via unitary order isomorphisms. We show that the ergodic decomposition of a quasi-regular Dirichlet space is unique up to a unique isomorphism of the indexing space. Furthermore, every unitary order isomorphism intertwining two quasi-regular Dirichlet spaces is decomposable over their ergodic decompositions up to conjugation via an isomorphism of the corresponding indexing spaces.

Let T T be a bijective nonsingular transformation on a finite measure space. We shall first construct a σ \sigma -finite and finite invariant measure by a unified method which is valid for both cases. Secondly we shall give another construction of a finite invariant measure. We shall also give a new necessary and sufficient condition of a unified form for the existence of σ \sigma -finite and finite invariant measures. Further, we shall discuss in detail ergodic transformations.

We study ergodic finite and infinite measures defined on the path space $X_{B}$ of a Bratteli diagram $B$ which are invariant with respect to the tail equivalence relation on $X_{B}$. Our interest is focused on measures supported by vertex and edge subdiagrams of $B$. We give several criteria when a finite invariant measure defined on the path space of a subdiagram of $B$ extends to a finite invariant measure on $B$. Given a finite ergodic measure on a Bratteli diagram $B$ and a subdiagram $B^{\prime }$ of $B$, we find the necessary and sufficient conditions under which the measure of the path space $X_{B^{\prime }}$ of $B^{\prime }$ is positive. For a class of Bratteli diagrams of finite rank, we determine when they have maximal possible number of ergodic invariant measures. The case of diagrams of rank two is completely studied. We also include an example which explicitly illustrates the proven results.

Interest in the study of dichotomy for various classes of probability measures arose after the well-known Kakutani theorem [1] concerning the equivalence and singularity of product measures had appeared. The development and various generalizations of this result can be found in Feldman [2], Hajek [3], Fernique [4], Kanter [5], and Okazaki [6]. In the last paper, Okazaki proved a dichotomy theorem for H-ergodic probability measures on a locally convex Hausdorff space equipped with a cylindrical c-algebra C(E, {x~ }), where {x*},~z C E*; this theorem generalizes all earlier dichotomy results, since in every case one can find an appropriate set H with respect to which the measures are ergodic. The proof of the Okasaki theorem uses topological properties of the original space and the separability of the c-algebra C(E, {x~ }) on which the measure p is defined. In this paper we prove a dichotomy theorem for ergodic measures on linear measure spaces in the most general situation under rather weak requirements. We do not assume that the linear space is equipped with a topological structure, and the measurable space structure need not agree with the linear operations. Let (E, ~) be an arbitrary measure space such that E is a linear set and the a-algebra ~ is invariant under translations by elements z E E. Let p be a probability measure on ~, and let H C E.

We classify the locally finite ergodic invariant measures of certain infinite interval exchange transformations (IETs). These transformations naturally arise from return maps of the straight-line flow on certain translation surfaces, and the study of the invariant measures for these IETs is equivalent to the study of invariant measures for the straight-line flow in some direction on these translation surfaces. For the surfaces and directions for which our methods apply, we can characterize the locally finite ergodic invariant measures of the straight-line flow in a set of directions of Hausdorff dimension larger than 1/2. We promote this characterization to a classification in some cases. For instance, when the surfaces admit a cocompact action by a nilpotent group, we prove each ergodic invariant measure for the straight-line flow is a Maharam measure, and we describe precisely which Maharam measures arise. When the surfaces under consideration are finite area, the straight-line flows in the directions we understand are uniquely ergodic. Our methods apply to translation surfaces admitting multi-twists in a pair of cylinder decompositions in non-parallel directions.

We consider the space of complete and separable metric spaces which are equipped with a probability measure. A notion of convergence is given based on the philosophy that a sequence of metric measure spaces converges if and only if all finite subspaces sampled from these spaces converge. This topology is metrized following Gromov’s idea of embedding two metric spaces isometrically into a common metric space combined with the Prohorov metric between probability measures on a fixed metric space. We show that for this topology convergence in distribution follows—provided the sequence is tight—from convergence of all randomly sampled finite subspaces. We give a characterization of tightness based on quantities which are reasonably easy to calculate. Subspaces of particular interest are the space of real trees and of ultra-metric spaces equipped with a probability measure. As an example we characterize convergence in distribution for the (ultra-)metric measure spaces given by the random genealogies of the Λ-coalescents. We show that the Λ-coalescent defines an infinite (random) metric measure space if and only if the so-called “dust-free”-property holds.

Chapter 27 - Measures on Algebraic—Topological Structures

Given Polish spaces X and Y and a Borel set S ⊆ X × Y with countable sections, we describe the circumstances under which a Borel function f : S → R is of the form f(x, y) = u(x) + v(y), where u : X → R and v : Y → R are Borel. This turns out to be a special case of the problem of determining whether a real-valued Borel cocycle on a countable Borel equivalence relation is a coboundary. We use several Glimm-Effros style dichotomies to give a solution to this problem in terms of certain σ-finite measures on the underlying space. The main new technical ingredient is a characterization of the existence of type III measures of a given cocycle. Suppose that S ⊆ X×Y and G is a group. A coordinatewise decomposition of a function f : S → G is a pair (u, v), where u : X → G, v : Y → G, and ∀(x, y) ∈ S (f(x, y) = u(x)v(y)). If X and Y are Polish spaces, G is a standard Borel group, and u and v are Borel, then we say that (u, v) is a Borel coordinatewise decomposition of f . Our main goal here is to show that when S is a Borel set with countable sections, f : S → G is Borel, and G = 〈R,+〉, the existence of a Borel coordinatewise decomposition can be characterized in terms of certain σ-finite measures on the disjoint union of X and Y (by a measure on a Polish space, we shall mean always a measure on its Borel subsets). Before getting to this, however, we consider first the existence of coordinatewise decompositions, without imposing any definability restrictions. ∗The author was supported in part by NSF VIGRE Grant DMS-0502315.