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We prove strengthenings of the Birkhoff Ergodic Theorem for weakly mixing and strongly mixing measure preserving systems. We show that our pointwise theorem for weakly mixing systems is strictly stronger than the Wiener-Wintner Theorem. We also show that our pointwise Theorems for weakly mixing and strongly mixing systems characterize weakly mixing systems and strongly mixing systems respectively. The methods of this paper also allow one to prove an enhanced pointwise ergodic theorem for other levels of the ergodic hierarchy such as ergodicity and mild mixing but not K-mixing. The author plans to include these additional pointwise ergodic theorems in his thesis., Comment: This paper is set to appear in Studia Mathematica

Extending a result by Chilin and Litvinov, we show by construction that given any \(\sigma \)-finite infinite measure space \((\Omega ,\mathcal {A}, \mu )\) and a function \(f\in L^1(\Omega )+L^\infty (\Omega )\) with \(\mu (\{|f|>\varepsilon \})=\infty \) for some \(\varepsilon >0\), there exists a Dunford–Schwartz operator T over \((\Omega ,\mathcal {A}, \mu )\) such that \(\frac{1}{N}\sum _{n=1}^N (T^nf)(x)\) fails to converge for almost every \(x\in \Omega \). In addition, for each operator we construct, the set of functions for which pointwise convergence fails almost everywhere is residual in \(L^1(\Omega )+L^\infty (\Omega )\).

In this paper, we establish a non-commutative analogue of Calderón’s transference principle, which allows us to deduce the non-commutative maximal ergodic inequalities from the special case—operator-valued maximal inequalities. As applications, we deduce the non-commutative Stein–Calderón maximal ergodic inequality and the dimension-free estimates of the non-commutative Wiener maximal ergodic inequality over Euclidean spaces. We also show the corresponding individual ergodic theorems. To show Wiener’s pointwise ergodic theorem, following a somewhat standard way we construct a dense subset on which pointwise convergence holds. To show Jones’ pointwise ergodic theorem, we use again the transference principle together with the Littlewood–Paley method, which is different from Jones’ original variational method that is still unavailable in the non-commutative setting.

A topic of classical interest in ergodic theory is extending Birkhoff's Pointwise Ergodic Theorem to various classes of nonconventional ergodic averages. Previous methods had only established the pointwise behavior of many such averages when applied to functions in L^p with p>1, leaving open the important case of L^1. In this thesis, I adapt and extend the techniques of Urban and Zienkiewicz and Buczolich and Mauldin, whose recent L^1 results have renewed interest in the subject, in order to prove several new results on the pointwise convergence of nonconventional averages of L^1 functions.The first result (Theorem 1.3) proves that averages along certain Bernoulli random sequences (generated by independent {0,1}-valued random variables) satisfy a pointwise ergodic theorem in L^1, with probability 1. The second (Theorem 1.4) proves that averages along n^d for d≥2, or along the sequence of primes, do not satisfy a pointwise ergodic theorem in L^1. (The result for n^2 was first shown by Buczolich and Mauldin; the others are new.) The third (Corollary 1.9) shows that, given a Fourier decay condition on a sequence of probability measures on the integers, some subsequence of the corresponding nonconventional averages will satisfy a pointwise ergodic theorem in L^1.

We show that if a σ-finite infinite measure space ( Ω , μ ) is quasi-non-atomic, then the Dunford–Schwartz pointwise ergodic theorem holds for f ∈ L 1 ( Ω ) + L ∞ ( Ω ) if and only if μ { | f | ≥ λ } ∞ for all λ > 0 .

Amos Nevo established the pointwise ergodic theorem in $L^p$ for measure-preserving actions of $\mathrm{PSL}_2(\mathbb{R})$ on probability spaces with respect to ball averages and every $p>1$. This paper shows by explicit example that Nevo's Theorem cannot be extended to $p=1$., 4 figures added

We establish a higher dimensional counterpart of Bourgain's pointwise ergodic theorem along an arbitrary integer-valued polynomial mapping. We achieve this by proving variational estimates $V_r$ on $L^p$ spaces for all $1\max\{p, p/(p-1)\}$. Moreover, we obtain the estimates which are uniform in the coefficients of a polynomial mapping of fixed degree.

International audience; An overview of last seven years results concerning Sarnak’s conjecture on Möbius disjointness is presented, focusing on ergodic theory aspects of the conjecture.

The Monte Carlo Method is a very useful and versatile numerical technique that allows to solve a large variety of problems difficult to tackle by other procedures. Even though the central idea is to simulate experiments on a computer and make inferences from the “observed” sample, it is applicable to problems that do not have an explicit random nature; it is enough if they have an adequate probabilistic approach. In fact, a frequent use of Monte Carlo techniques is the evaluation of definite integrals that at first sight have no statistical nature but can be interpreted as expected values under some distribution. In this lecture we shall present and justify essentially all the procedures that are commonly used in particle physics and statistics leaving aside subjects like Markov Chains that deserve a whole lecture by themselves and for which only the relevant properties will be stated without demonstration.

In this note we give a short proof of a pointwise ergodic theorem for measure-preserving actions of word hyperbolic groups, also obtained recently by Bufetov, Khristoforov and Klimenko. Our approach also applies to infinite measure spaces, and one application is to linear actions of discrete groups on the plane.

In this note we prove the a pointwise ergodic theorem for functions taking values in a separable complete CAT(0)-space, analogous to Lindenstrauss' pointwise ergodic theorem for real-valued integrable functions on a probability space subject to a probability-preserving action of an amenable l.c.s.c. group, where in the CAT(0) setting the role of ergodic averages is played by the barycentres of the empirical distributions of a CAT(0)-valued map along an orbit of the group action. The proof rests on an approximation argument and an appeal to that result for real-valued maps., 7 pages; [TDA, April 12st 2011]: Modified slightly following referee reports, and a small mistake corrected; [v5:] This preprint has been re-written to correct to a mistake in the proof of Lemma 2.3. The journal published that correction in a separate erratum

We present a modified version of Buczolich and Mauldin’s proof that the sequence of square numbers is universally L1-bad. We extend this result to a large class of sequences, including the dth powers and the set of primes. Furthermore, we show that any subsequence of the averages taken along these sequences is also universally L1-bad.

Motivated by the variations of Sarnak's conjecture due to El Abdalaoui, Kulaga-Przymus, Lemanczyk, De La Rue and by the observation that the Mobius function is a good weight (with limit zero) for the polynomial pointwise ergodic theorem in $L^q$, q>1, we introduce polynomial versions of the Sarnak conjecture for minimal systems., 5 pages. The formulation of Theorem 2.2 has been corrected

Let $a_n$ be the random increasing sequence of natural numbers which takes each value independently with probability $n^{-a}$, $0 < a < 1/2$, and let $p(n) = n^{1+\epsilon}$, $0 < \epsilon < 1$. We prove that, almost surely, for every measure-preserving system $(X,T)$ and every $f \in L^1(X)$ the modulated, random averages \[ \frac{1}{N} \sum_{n = 1}^N e(p(n)) T^{a_n(\omega)} f\] converge to $0$ pointwise almost everywhere., Comment: v2: corrected the chain of approximations, see (2.8) in v2; small corrections following referee report

We prove that for any bounded functions $f_1, f_2$ on a measure-preserving dynamical system $(X,T)$ and any distinct integers $a_1, a_2$, for almost every $x$ the sequence $$ f_1(T^{a_1 n}x) f_2(T^{a_2 n}x) $$ is a good weight for the pointwise ergodic theorem., Comment: v2: 8 pages, improved typography

As we saw in previous chapters, ergodic theorems, even though being originally motivated by a recurrence question from physics, found applications in unexpected areas such as number theory. So it is not surprising that they attracted continuous attention among the mathematical community, thus leading to various generalizations and extensions. In this chapter we describe a very few of them.

Let T be a positive L1-L∞ contraction. We prove that the following statements are equivalent in constructive mathematics.(1) The projection in L2, on the space of invariant functions exists:(2) The sequence (Tn)n∈N Cesáro-converges in the L2 norm:(3) The sequence (Tn)n∈N Cesáro-converges almost everywhere.Thus, we find necessary and sufficient conditions for the Mean Ergodic Theorem and the Dunford-Schwartz Pointwise Ergodic Theorem.As a corollary we obtain a constructive ergodic theorem for ergodic measure-preserving transformations.This answers a question posed by Bishop.

Generalizing the one-parameter case, we prove that the orbit of a point on a compact nilmanifold X under a polynomial action of $\mathbb{Z}^{d}$ by translations on X is uniformly distributed on the union of several sub-nilmanifolds of X . As a corollary we obtain the pointwise ergodic theorem for polynomial actions of $\mathbb{Z}^{d}$ by translations on a nilmanifold.

First, we show that there exists a sequence $(a_n)$ of integers which is a good averaging sequence in $L^2$ for the pointwise ergodic theorem and satisfies $$ \frac{a_{n+1}}{a_n}>e^{(\log n)^{-1-\epsilon}} $$ for $n>n(\epsilon)$. This should be contrasted with an earlier result of ours which says that if a sequence $(a_n)$ of integers (or real numbers) satisfies $$ \frac{a_{n+1}}{a_n}>e^{(\log n)^{-\frac{1}{2}+\epsilon}} $$ for some positive $\epsilon$, then it is a bad averaging sequence in $L^2$ for the pointwise ergodic theorem.Another result of the paper says that if we select each integer $n$ with probability $1/n$ into a random sequence, then, with probability 1, the random sequence is a bad averaging sequence for the mean ergodic theorem. This result should be contrasted with Bourgain's result which says that if we select each integer $n$ with probability $\sigma_n$ into a random sequence, where the sequence $(\sigma_n)$ is decreasing and satisfies $$ \lim_{t\to\infty}\frac{\sum_{n\le t}\sigma_n}{\log t}=\infty, $$ then, with probability 1, the random sequence is a good averaging sequence for the mean ergodic theorem.

In this paper we prove the pointwise ergodic theorem for general locally compact amenable groups along Folner sequences that obey some restrictions. These restrictions are mild enough so that such sequences exist for all amenable groups. We also prove a generalization of the Shannon-McMillan-Breiman theorem to all discrete amenable groups. -->