Chapter 27 - Measures on Algebraic—Topological Structures

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.

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

DOROTHY MAHARAMC) 0. Introduction.0.1.Suppose we are given a nonsingular measurable one-one transformation T of a measure space (X, m) onto itself.The main object of this paper is to answer the question : Under what conditions will there exist a measure p (equivalent to m) which is invariant under T (that is, with respect to which T becomes measurepreserving)(2) ?We shall see that this question can be answered, in principle, if one merely knows the sequence of Radon-Nikodym derivatives con(x) = dm(Tnx)/dm(x) ( = 1,2,...).We obtain necessary and sufficient conditions for the existence of an invariant measure in terms of these derivatives (see Theorem 7, in 5.1 below).We also obtain a construction for an invariant measure, when one exists; and we discuss a few other questions which arise in the course of the investigation.All measures considered are to be a-finite and equivalent to m ; and, throughout almost all the paper, we make the restriction that the given measure m is finite.This involves no real loss of generality ; and in 6.1 we discuss briefly the effect of eliminating this restriction.But the finiteness of m does enable most of our results to be stated much more simply.In the case of a finite invariant measure, the situation is relatively simple ; in Theorem 1 (3.2) a number of criteria, not all of them new, are given.Perhaps the simplest of them (essentially known) is that a finite invariant measure exists if and only if the (C, 1) limit l(x) of the sequence {con(x)} exists and is positive a.e. ; further, if this condition is fulfilled, J" / dm gives a finite invariant measure.The case of an infinite invariant measure is decidedly more complicated.The main idea is to average the sequence {con(x)} in a more delicate way, by introducing the notion of a "density average".For any sufficiently "smooth" nonnegative sequence {()} and for all a, ^O, we consider the relative asymptotic density of the 's for which ()^ a as compared with those for which (n)^, and integrate this with respect to a to get the "density average relative to ".It turns out that, when applied to the sequence {con(x)} with = 0, this process produces the (C, 1) limit and thus gives a criterion and a simple construction for a finite invariant measure.

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 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.

About fifty years ago, questions on the existence and non-existence of finite invariant\nmeasures were studied by various authors and from different directions. In this article,\nwe examine these classical results and prove directly that all the conditions introduced\nby these authors are equivalent to each other. We begin at the fundamental level of a\nrecurrent transformation whose properties can be strengthened to obtain the aforementioned\nclassical results for the existence of a finite invariant measure. We conclude with the\nintroduction of a new property, Strongly Weakly Wandering (sww) sequences, the existence\nof which is equivalent to the non-existence of a finite invariant measure. It is shown\nthat every sww sequence is also an Exhaustive Weakly Wandering (eww) sequence for ergodic\ntransformations. Although all ergodic transformations with no finite invariant measure are\nknown to have eww sequences, there are exceedingly few actual examples for which explicit\neww sequences can be exhibited. The significance of sww sequences is that it provides a\ncondition which is easier to verify than the condition for eww sequences (Proposition\n4.5). In a second paper, we will continue these studies and also connect them to some of\nthe more recent derived conditions for finite invariant measures. The impetus for this\nwork, began with the late Professor Shizuo Kakutani, with whom the authors worked and had\nmany fruitful discussions on these topics.

We discuss conditions for the existence of invariant measures of smooth dynamical systems on compact manifolds. If there is an invariant measure with continuously differentiable density, then the divergence of the vector field along every solution tends to zero in the Cesàro sense as time increases unboundedly. Here the Cesàro convergence may be replaced, for example, by any Riesz summation method, which can be arbitrarily close to ordinary convergence (but does not coincide with it). We give an example of a system whose divergence tends to zero in the ordinary sense but none of its invariant measures is absolutely continuous with respect to the `standard' Lebesgue measure (generated by some Riemannian metric) on the phase space. We give examples of analytic systems of differential equations on analytic phase spaces admitting invariant measures of any prescribed smoothness (including a measure with integrable density), but having no invariant measures with positive continuous densities. We give a new proof of the classical Bogolyubov–Krylov theorem using generalized functions and the Hahn–Banach theorem. The properties of signed invariant measures are also discussed.

To infer eigenvalues of the infinite-dimensional Koopman operator, we study the leading eigenvalues of the autocovariance matrix associated with a given observable of a dynamical system. For any observable f for which all the time-delayed autocovariance exist, we construct a Hilbert space H_{f} and a Koopman-like operator K that acts on H_{f}. We prove that the leading eigenvalues of the autocovariance matrix has one-to-one correspondence with the energy of f that is represented by the eigenvectors of K. The proof is associated to several representation theorems of isometric operators on a Hilbert space, and the weak-mixing property of the observables represented by the continuous spectrum. We also provide an alternative proof of the weakly mixing property. When f is an observable of an ergodic dynamical system which has a finite invariant measure μ,H_{f} coincides with closure in L^{2}(X,dμ) of Krylov subspace generated by f, and K coincides with the classical Koopman operator. The main theorem sheds light to the theoretical foundation of several semi-empirical methods, including singular spectrum analysis (SSA), data-adaptive harmonic analysis (DAHD), Hankel DMD, and Hankel alternative view of Koopman analysis (HAVOK). It shows that, when the system is ergodic and has finite invariant measure, the leading temporal empirical orthogonal functions indeed correspond to the Koopman eigenfrequencies. A theorem-based practical methodology is then proposed to identify the eigenfrequencies of K from a given time series. It builds on the fact that the convergence of the renormalized eigenvalues of the Gram matrix is a necessary and sufficient condition for the existence of K-eigenfrequencies. Numerical illustrating results on simple low dimensional systems and real interpolated ocean sea-surface height data are presented and discussed.

It is shown that every $\sigma$-additive $\sigma$-finite invariant measure on an abelian group has a proper $\sigma$-additive invariant extension.

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.

This book contains a collection of survey papers by leading researchers in ergodic theory, low-dimensional and topological dynamics and it comprises nine chapters on a range of important topics. These include: the role and usefulness of ultrafilters in ergodic theory, topological dynamics and Ramsey theory; topological aspects of kneading theory together with an analogous 2-dimensional theory called pruning; the dynamics of Markov odometers, Bratteli-Vershik diagrams and orbit equivalence of non-singular automorphisms; geometric proofs of Mather's connecting and accelerating theorems; recent results in one dimensional smooth dynamics; periodic points of nonexpansive maps; arithmetic dynamics; the defect of factor maps; entropy theory for actions of countable amenable groups.

In Chapter 10 we showed how a topological dynamical system \( (K;\varphi) \) gives rise to a measure-preserving system by choosing a \( \varphi \)-invariant measure on K. The existence of such a measure is guaranteed by the Krylov–Bogoljubov Theorem 10.2. In general there can be many invariant measures, and we also investigated how minimality of the topological system is reflected in properties of the associated measure-preserving system. It is now our aim to go in the other direction: starting from a given measure-preserving system we shall construct some topological system (sometimes called topological model) and an invariant measure so that the resulting measure-preserving system is isomorphic to the original one. By doing this, methods from the theory of topological dynamical system will be available, and we may gain further insights into measure-preserving systems. Thus, by switching back and forth between the measure theoretic and the topological situation, we can deepen our understanding of dynamical systems. In particular, this procedure will be carried out in Chapter 17

It is shown that for the vector space $\mathbb{R^N}$ (equipped with the product topology and the Yamasaki-Kharazishvili measure) the group of linear measure preserving isomorphisms is quite rich. Using Kharazishvili's approach, we prove that every infinite-dimensional Polish linear space admits a $\sigma$-finite non-trivial Borel measure that is translation invariant with respect to a dense linear subspace. This extends a~recent result of Gill, Pantsulaia and Zachary on the existence of such measures in Banach spaces with Schauder bases. It is shown that each $\sigma$-finite Borel measure defined on an infinite-dimensional Polish linear space, which assigns the value 1 to a fixed compact set and is translation invariant with respect to a~linear subspace fails the uniqueness property. For Banach spaces with absolutely convergent Markushevich bases, a similar problem for the usual completion of the concrete $\sigma$-finite Borel measure is solved positively. The uniqueness problem for non-$\sigma$-finite semi-finite translation invariant Borel measures on a Banach space $X$ which assign the value 1 to the standard rectangle (i.e., the rectangle generated by an absolutely convergent Markushevich basis) is solved negatively. In addition, it is constructed an example of such a measure $\mu_B^0$ on $X$, which possesses a strict uniqueness property in the class of all translation invariant measures which are defined on the domain of $\mu_B^0$ and whose values on non-degenerate rectangles coincide with their volumes.

We shortly mention several perturbation problems of classical dynamical systems by stochastic forces. We look more closely to the case of an Hamiltonian system consisting of a particle moving in R d under the action of a force derived from a potential V and an additional stochastic force. We report on a recent extension of ours with Zehnder of results by Potter, Mc Kean, Markus and Weerasinghe. Under restrictions on the growth of V at infinity or attractiveness of the force towards the origin we give existence, uniqueness and stability results for the solution of the (stochastic) equations of motion. We also give a comparison theorem with solutions of a corresponding linearised system, via a Cameron-Martin-Girsanov-Maruyama type of formula. We also discuss the asymptotic behavior of the solution for large times, as well as the existence of a σ-finite, not finite invariant measure, the Lebesgue measure in phase space.

In Section I.1 we highlighted the fact that some maps have an invariant measure naturally associated with them, but that ergodic theory also comprises, among its aims, the dynamical study of maps which are not born with an associated invariant measure. The first step in this direction was showing that every continuous map of a compact metric space has an invariant measure (I.8). The second was the ergodic decomposition theorem and the concept of regular points (II.6). Not much more can be said about continuous maps in general; in order to develop our theory further, we will restrict our attention to differentiable maps of closed manifolds. Before doing that, however, we will introduce the concept of expanding maps of metric spaces—roughly, maps which locally increase distances. For such maps, it is possible to find, among all invariant measures, one whose properties make it especially interesting. We start with two particular cases of the definition, where the results obtained are particularly relevant.