Topological Wiener–Wintner ergodic theorems and a randomL2ergodic theorem

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 initiate the study of effective pointwise ergodic theorems in resource-bounded settings. Classically, the convergence of the ergodic averages for integrable functions can be arbitrarily slow [Ulrich Krengel, 1978]. In contrast, we show that for a class of PSPACE L¹ functions, and a class of PSPACE computable measure-preserving ergodic transformations, the ergodic average exists and is equal to the space average on every EXP random. We establish a partial converse that PSPACE non-randomness can be characterized as non-convergence of ergodic averages. Further, we prove that there is a class of resource-bounded randoms, viz. SUBEXP-space randoms, on which the corresponding ergodic theorem has an exact converse - a point x is SUBEXP-space random if and only if the corresponding effective ergodic theorem holds for x.

We establish results with an arithmetic flavor that generalize the polynomial multidimensional Szemerédi theorem and related multiple recurrence and convergence results in ergodic theory. For instance, we show that in all these statements we can restrict the implicit parameter n n to those integers that have an even number of distinct prime factors or satisfy any other congruence condition. In order to obtain these refinements we study the limiting behavior of some closely related multiple ergodic averages with weights given by appropriately chosen multiplicative functions. These averages are then analyzed using a recent structural result for bounded multiplicative functions proved by the authors.

We investigate the existence, properties and approach to stationary non-equilibrium states of infinite harmonic crystals. For classical systems these stationary states are, like the Gibbs states, Gaussian measures on the phase space of the infinite system (analogues results are true for quantum systems). Their ergodic properties are the same as those of the equilibrium states: e.g. for ordered periodic crystals they are Bernoulli. Unlike the equilibrium states however they are not “stable” towards perturbations in the potential. We are particularly concerned here with states in which there is a non-vanishing steady heat flux passing through “every point” of the infinite system. Such “superheat-conducting” states are of course only possible in systems in which Fourier's law does not hold: the perfect harmonic crystal being an example of such a system. For a one dimensional system, we find such states (explicitely) as limits, whent→∞, of time evolved initial states μ i in which the “left” and “right” parts of the infinite crystal are in “equilibrium” at different temperatures, β − ≠β −1 , and the “middle” part is in an arbitrary state. We also investigate the limit of these stationary (t→∞) states as the coupling strength λ between the “system” and the “reservoirs” goes to zero. In this limit we obtain a product state, where the reservoirs are in equilibrium at temperatures β −1 and β −1 and the system is in the unique stationary state of the reduced dynamics in the weak coupling limit.

The soliton resolution conjecture for the focusing nonlinear Schrödinger equation (NLS) is the vaguely worded claim that a global solution of the NLS, for generic initial data, will eventually resolve into a radiation component that disperses like a linear solution, plus a localized component that behaves like a soliton or multisoliton solution. Considered to be one of the fundamental open problems in the area of nonlinear dispersive equations, this conjecture has eluded a proof or even a precise formulation to date.This paper proves a “statistical version” of this conjecture at mass‐subcritical nonlinearity, in the following sense: The uniform probability distribution on the set of all functions with a given mass and energy, if such a thing existed, would be a natural invariant measure for the NLS flow and would reflect the long‐term behavior for “generic initial data” with that mass and energy. Unfortunately, such a probability measure does not exist. We circumvent this problem by constructing a sequence of discrete measures that, in principle, approximate this fictitious probability distribution as the grid size goes to 0. We then show that a continuum limit of this sequence of probability measures does exist in a certain sense, and in agreement with the soliton resolution conjecture, the limit measure concentrates on the unique ground state soliton. Combining this with results from ergodic theory, we present a tentative formulation and proof of the soliton resolution conjecture in the discrete setting.The above results, following in the footsteps of a program of studying the long‐term behavior of nonlinear dispersive equations through their natural invariant measures initiated by Lebowitz, Rose, and Speer and carried forward by Bourgain, McKean, Tzvetkov, Oh, and others, are proved using a combination of techniques from large deviations, PDEs, harmonic analysis, and bare‐hands probability theory. It is valid in any dimension. © 2014 Wiley Periodicals, Inc.

Karlsson and Margulis (Commun. Math. Phys. 208:107–123, 1999) proved in the setting of uniformly convex geodesic spaces, which additionally satisfy a nonpositive curvature condition, an ergodic theorem that focuses on the asymptotic behavior of integrable cocycles of nonexpansive mappings over an ergodic measure-preserving transformation. In this note we show that this result holds true when assuming a weaker notion of uniform convexity.

Generalizations and refinements are given for results of Kozlov and Treschev on non-uniform averagings in the ergodic theorem in the case of operator semigroups on spaces of integrable functions and semigroups of measure-preserving transformations. Conditions on the averaging measures are studied under which the averages converge for broad classes of integrable functions. Bibliography: 96 items.

We study the one parameter family of potential functions associated with the unstable Jacobian potential (or geometric potential) for the geodesic flow of a compact rank 1 surface of nonpositive curvature. For q<1, it is known that there is a unique equilibrium state associated with , and it has full support. For q>1 it is known that an invariant measure is an equilibrium state if and only if it is supported on the singular set. We study the critical value q = 1 and show that the ergodic equilibrium states are either the restriction to the regular set of the Liouville measure or measures supported on the singular set. In particular, when q = 1, there is a unique ergodic equilibrium state that gives positive measure to the regular set.

In this paper) we consider an arbitrary irreducible random walk and an ar~ bitrary stationary and ergodic random scenery on Zd. \Ve find conditions on their distributions such that the associated random walk in random scenery is or is not weak Bernoulli. Our results extend an earlier das~ sification for the special case in which the individual scenery values arc assumed to be independent and identically distributed random variables. 1 Background and main theorems 1.1 Background. For a fixed integer d ~ I, let X = (Xn)nE:J: be independent and identically distributed random variables taking values in Zd according to a common distribution 't11 OIl Z d satisfying (*1) m is irreducible, i.e.) supp(m) = {;r;: m(;r;) > O} generates the group Zd. Let = be the corresponding random walk on Z d, defined by 5 = 0, l\eA'i, let C = be a random scener,1J on Zd taking values in a finite set 'yd F according to a distribution ft on F J • satisfying (*2) ft is stationary with respect to translations in the group Z d; (*3) ft is ergodic with respect to translations in fm, the subgroup of Zd generated by SUpp(lII) SUpp(lII) = { 11' 111()111(11) > O}. The random walk and the random scenery are assumed to be independent. The joint process y= with }~, = (Xn, Csl is called the random walk m random scener,1J associated with m and ft. In ergodic theory, }' is an example of what is called a skew prod'uct. It is obvious that, because of (*2), }' is a stationary process. The role of (*3) is to make }' non-periodic. Indeed, if F = {O, I}, d = I, m is simple random ~ ReKearch :mppoJ1ed in paJ1 by KBN gran1 2 PO:JA 04G22 (2002-2005) j' ReKearch :mppoJ1ed in paJ1 by a gran1 from 1he S\vedh,h Na1 ural Science ReKearch Council and by NSF-gran1 DfvlS-010:JS41.

This book provides a detailed introduction to the ergodic theory of equilibrium states giving equal weight to two of its most important applications, namely to equilibrium statistical mechanics on lattices and to (time discrete) dynamical systems. It starts with a chapter on equilibrium states on finite probability spaces which introduces the main examples for the theory on an elementary level. After two chapters on abstract ergodic theory and entropy, equilibrium states and variational principles on compact metric spaces are introduced emphasizing their convex geometric interpretation. Stationary Gibbs measures, large deviations, the Ising model with external field, Markov measures, Sinai-Bowen-Ruelle measures for interval maps and dimension maximal measures for iterated function systems are the topics to which the general theory is applied in the last part of the book. The text is self contained except for some measure theoretic prerequisites which are listed (with references to the literature) in an appendix.

Kingman’s subadditive ergodic theorem is traditionally proved in the setting of a measure-preserving invertible transformationTof a measure space$(X, \mu )$. We use a theorem of Silva and Thieullen to extend the theorem to the setting of a not necessarily invertible transformation, which is non-singular under the assumption that$\mu $and$\mu \circ T$have the same null sets. Using this, we are able to produce versions of the Furstenberg–Kesten theorem and the Oseledeč ergodic theorem for products of random matrices without the assumption that the transformation is either invertible or measure-preserving.

When the Integral is the Limit

Furstenberg–Zimmer structure theory refers to the extension of the dichotomy between the compact and weakly mixing parts of a measure-preserving dynamical system and the algebraic and geometric descriptions of such parts to a conditional setting, where such dichotomy is established relative to a factor and conditional analogs of those algebraic and geometric descriptions are sought. Although the unconditional dichotomy and the characterizations are known for arbitrary systems, the relative situation is understood under certain countability and separability hypotheses on the underlying groups and spaces. The aim of this article is to remove these restrictions in the relative situation and establish a Furstenberg–Zimmer structure theory in full generality. As an independent byproduct, we establish a connection between the relative analysis of systems in ergodic theory and the internal logic in certain Boolean topoi.

This workshop in real and harmonic analysis surveyed recent investigations in areas including geometric measure theory and restriction theory, multiple ergodic averages, local smoothing estimates, Schrödinger and Hörmander-type oscillatory integral operators, multilinear estimates and analysis on the Hamming cube. In particular, the workshop emphasized the recent solutions of two longstanding problems: the Kakeya phenomenon in dimension three, and almost everywhere convergence of long time averages associated to multiple commuting measure-preserving transformations. The methods presented during the workshop have yielded applications in ergodic theory, number theory, and computer science.

0. Summary. The objects of ergodic theory -measure spaces with measure-preserving transformation groups-wil l be called processes, those of topological dynamics-compact metric spaces with groups of homeomorphisms-will be called flows. We shall be concerned with what may be termed the "arithmetic" of these classes of objects. One may form products of processes and of flows, and one may also speak of factor processes and factor flows. By analogy with the integers, we may say that two processes are relatively prime if they have no non-trivial factors in common. An alternative condition is that whenever the two processes appear as factors of a third process, then their product too appears as a factor. In our theories it is unknown whether these two conditions are equivalent. We choose the second of these conditions as the more useful and refer to it as disjointness. Our first applications of the concept of disjointness are to the classification of processes and flows. It will appear that certain classes of processes (flows) may be characterized by the property of being disjoint from the members of other classes of processes (flows). For example the processes with entropy 0 are just those which are disjoint from all Bernoulli flows. Another application of disjointness of processes is to the following filtering problem. If {xn} and {Yn} represent two stationary stochastic processes, when can {xn} be filtered perfectly from {Xn + Yn}? We will find (Part I, §9) that a sufficient condition is the disjointness of the processes in question. For flows the principal application of disjointness is to the ~tudy of properties of minimal sets (Part III). Consider the flow on the unit circle K = {z: [zl = 1 } that arises from the transformation z --~ z 2. What can be said about the "size" of the minimal sets for this flow, that is, closed subsets of K invariant under z ~ z ~, but not containing proper subsets with these properties. Uncountably many such minimal sets exist in K. Writing z = exp (2~ri Ean/2n), an = 0, 1, we see that this amounts to studying the mini-