Ergodic Theorems and Converses for PSPACE Functions

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

Let$m\in \mathbb{N}$and$\mathbf{X}=(X,{\mathcal{X}},\unicode[STIX]{x1D707},(T_{\unicode[STIX]{x1D6FC}})_{\unicode[STIX]{x1D6FC}\in \mathbb{R}^{m}})$be a measure-preserving system with an$\mathbb{R}^{m}$-action. We say that a Borel measure$\unicode[STIX]{x1D708}$on$\mathbb{R}^{m}$is weakly equidistributed for$\mathbf{X}$if there exists$A\subseteq \mathbb{R}$of density 1 such that, for all$f\in L^{\infty }(\unicode[STIX]{x1D707})$, we have$$\begin{eqnarray}\lim _{t\in A,t\rightarrow \infty }\int _{\mathbb{R}^{m}}f(T_{t\unicode[STIX]{x1D6FC}}x)\,d\unicode[STIX]{x1D708}(\unicode[STIX]{x1D6FC})=\int _{X}f\,d\unicode[STIX]{x1D707}\end{eqnarray}$$for$\unicode[STIX]{x1D707}$-almost every$x\in X$. Let$W(\mathbf{X})$denote the collection of all$\unicode[STIX]{x1D6FC}\in \mathbb{R}^{m}$such that the$\mathbb{R}$-action$(T_{t\unicode[STIX]{x1D6FC}})_{t\in \mathbb{R}}$is not ergodic. Under the assumption of the pointwise convergence of the double Birkhoff ergodic average, we show that a Borel measure$\unicode[STIX]{x1D708}$on$\mathbb{R}^{m}$is weakly equidistributed for an ergodic system$\mathbf{X}$if and only if$\unicode[STIX]{x1D708}(W(\mathbf{X})+\unicode[STIX]{x1D6FD})=0$for every$\unicode[STIX]{x1D6FD}\in \mathbb{R}^{m}$. Under the same assumption, we also show that$\unicode[STIX]{x1D708}$is weakly equidistributed for all ergodic measure-preserving systems with$\mathbb{R}^{m}$-actions if and only if$\unicode[STIX]{x1D708}(\ell )=0$for all hyperplanes$\ell$of $\mathbb{R}^{m}$. Unlike many equidistribution results in literature whose proofs use methods from harmonic analysis, our results adopt a purely ergodic-theoretic approach.

We study the arithmetic analogue of maximal functions on diagonal hypersurfaces. This paper is a natural step following the papers of [13], [14] and [16]. We combine more precise knowledge of oscillatory integrals and exponential sums to generalize the asymptotic formula in Waring’s problem to an approximation formula for the Fourier transform of the solution set of lattice points on hypersurfaces arising in Waring’s problem and apply this result to arithmetic maximal functions and ergodic averages. In sufficiently large dimensions, the approximation formula, l2-maximal theorems and ergodic theorems were previously known. Our contribution is in reducing the dimensional constraint in the approximation formula using recent bounds of Wooley, and improving the range of lp spaces in the maximal and ergodic theorems. We also conjecture the expected range of spaces.

We use a non-commutative generalization of the Banach Principle to show that the classical individual ergodic theorem for subsequences generated by means of uniform sequences can be extended to the von Neumann algebra setting.

The rate of convergence in von Neumann's mean ergodic theorem is studied for continuous time. The condition that the rate of convergence of the ergodic averages be of power-law type is shown to be equivalent to requiring that the spectral measure of the corresponding dynamical system have a power-type singularity at 0. This forces the estimates for the convergence rate in the above ergodic theorem to be necessarily spectral. All the results obtained have obvious exact analogues for wide-sense stationary processes.Bibliography: 7 titles.

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.

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.

The Jewett–Krieger theorem states that each ergodic system has a strictly ergodic topological model. In this article, we show that for an ergodic system one may require more properties on its strictly ergodic model. For example, the orbit closure of points in diagonal under face transforms may be also strictly ergodic. As an application, we show the pointwise convergence of ergodic averages along cubes, which was firstly proved by Assani (J Anal Math 110:241–269, 2010).

We provide an explicit uniform bound on the local stability of ergodic averages in uniformly convex Banach spaces. Our result can also be viewed as a finitary version in the sense of Tao of the mean ergodic theorem for such spaces and so generalizes similar results obtained for Hilbert spaces by Avigad et al [Local stability of ergodic averages. Trans. Amer. Math. Soc. to appear] and Tao [Norm convergence of multiple ergodic averages for commuting transformations. Ergod. Th. & Dynam. Sys.28(2) (2008), 657–688].

We study double averages along orbits for measure-preserving actions of$\mathbb{A}^{{\it\omega}}$, the direct sum of countably many copies of a finite abelian group$\mathbb{A}$. We show an$\text{L}^{p}$norm-variation estimate for these averages, which in particular re-proves their convergence in$\text{L}^{p}$for any finite$p$and for any choice of two$\text{L}^{\infty }$functions. The result is motivated by recent questions on quantifying convergence of multiple ergodic averages.

The notion of ergodicity of a measure-preserving transformation is generalized to finite sets of transformations. The main result is that ifT1,T2, …,Ts are invertible commuting measure-preserving transformations of a probability space (X, ℬ, μ) then $$\frac{1}{{N - M}}\sum\limits_{n = M}^{N - 1} {T{}_1^n } f_1 .T_2^n f_2 .....T_s^n f_s \xrightarrow[{N - M \to \propto }]{{I^2 (X)}}(\int_X {f1d\mu )} (\int_X {f2d\mu )...(\int_X {fsd\mu )} } $$ (1) for anyf1,f2, …,fs∈Lx(X, ℬ, μ) iffT1×T2×…×Ts and all the transformationsTiTj1,i≠j, are ergodic. The multiple recurrence theorem for a weakly mixing transformation follows as a special case.

We give some topological ergodic theorems inspired by theWiener-Wintnerergodic theorem. These theorems are used to give results for uniquely ergodic transformations and to study unique equilibrium states for shift maps. These latter results give randomL2ergodic theorems for a finite set of commuting measure-preserving transformations.

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.

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.

Using a version of an ergodic lemma due to Cuculescu and Foiaş, we prove a pointwise ergodic theorem for L 1 L^1 -contractions which can be viewed as a perturbed version of the celebrated ergodic theorem of Chacon and Ornstein. Surprisingly, to some extent, the complex part of the iterates involved have no effect on the ergodic convergence.