Let T1,..., Tl: X → X be commuting measure-preserving transformations on a probability space (X, X, µ). We show that the multiple ergodic averages 1 PN−1 N n=0 f1(T n 1 x)... fl(T n l x) are convergent in L2 (X, X, µ) as N → ∞ for all f1,..., fl ∈ L ∞ (X, X, µ); this was previously established for l = 2 by Conze and Lesigne [2] and for general l assuming some additional ergodicity hypotheses on the maps Ti and TiT −1 j by Frantzikinakis and Kra [3] (with the l = 3 case of this result established earlier in [29]). Our approach is combinatorial and finitary in nature, inspired by recent developments regarding the hypergraph regularity and removal lemmas, although we will not need the full strength of those lemmas. In particular, the l = 2 case of our arguments are a finitary analogue of those in [2].

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(Communicated by Carlangelo Liverani) Abstract. We consider the question of computing invariant measures from an abstract point of view. Here, computing a measure means finding an algorithm which can output descriptions of the measure up to any precision. We work in a general framework (computable metric spaces) where this problem can be posed precisely. We will find invariant measures as fixed points of the transfer operator. In this case, a general result ensures the computability of isolated fixed points of a computable map. We give general conditions under which the transfer operator is computable on a suitable set. This implies the computability of many “regular enough” invariant measures and among them many physical measures. On the other hand, not all computable dynamical systems have a computable invariant measure. We exhibit two examples of computable dynamics, one having a physical measure which is not computable and one for which no invariant measure is computable, showing some subtlety in this kind of problems. 1. Introduction. An

Abstract This paper offers some new results on randomness with respect to classes of measures, along with a didactical exposition of their context based on results that appeared elsewhere. We start with the reformulation of the Martin-Löf definition of randomness (with respect to computable measures) in terms of randomness deficiency functions. A formula that expresses the randomness deficiency in terms of prefix complexity is given (in two forms). Some approaches that go in another direction (from deficiency to complexity) are considered. The notion of Bernoulli randomness (independent coin tosses for an asymmetric coin with some probability p of head) is defined. It is shown that a sequence is Bernoulli if it is random with respect to some Bernoulli

In this paper, we present some aspects of a recent application of proof mining by J.Avigad, H.Towsner and the author. In this case study, we analysed a proof of the Mean Ergodic Theorem and obtained a computable rate of convergence for the ergodic averages. Proof mining generally falls into two main categories: Establishing general metatheorems that classify theorems and proofs from which additional information may be extracted and carrying out case studies. The aim of presenting aspects of a proof analysis in detail in this paper is to illustrate how the general logical results and the techniques they rely on translate into a proof analysis in practice. 1

Abstract Let X1, X2,... be a sequence of identically distributed, pairwise independent random variables with distribution P. Let the expected value be µ < ∞. Let S n = ∑ n i=1 Xi. It is well-known that S n/n converges to µ almost surely. We show that this convergence is effective in (P,µ). In particular, if P,µ are computable then the convergence is effective. On the other hand, if the convergence is effective in P then µ is computable from P. The effectiveness of convergence is detached in the sense that nothing can be inferred about the speed of convergence in the law of large numbers from the speed of convergence in computing P and µ. This theorem can be used to show an effective renewal theorem, which then can be used to prove an effective ergodic theorem for countable Markov chains. The last result is a special case of effective ergodic theorems proven by Avigad-Gerhardy-Towsner and Galatolo-Hoyrup-Rojas, but we hope that the direct constructivization of the probability-theory proofs is still useful. 1

Abstract. We exhibit a close correspondence between L1-computable functions and Schnorr tests. Using this correspondence, we prove that a point x ∈ [0,1] d is Schnorr random if and only if the Lebesgue Differentiation Theorem holds at x for all L1-computable functions f ∈ L1([0,1] d). 1.

Dedicated to Grigori Mints in honor of his seventieth birthday. Almost from the inception of Hilbert’s program, foundational and structural efforts in proof theory have been directed towards the goal of clarifying the computational content of modern mathematical methods. This essay surveys various methods of extracting computational information from proofs in classical first-order arithmetic, and reflects on some of the relationships between them. Variants of the Gödel-Gentzen doublenegation translation, some not so well known, serve to provide canonical and efficient computational interpretations of that theory. 1

In this survey we present some recent applications of proof mining to the fixed point theory of (asymptotically) nonexpansive mappings and to the metastability (in the sense of Terence Tao) of ergodic averages in uniformly convex Banach spaces. Contents 1 Proof Mining 2 2 Some topics in fixed point theory of nonexpansive mappings 3 2.1 The approximate fixed point property......................... 4 2.2 Krasnoselski-Mann iterations.............................. 6

Abstract. According to the Furstenberg-Zimmer structure theorem, every measure-preserving system has a maximal distal factor, and is weak mixing relative to that factor. Furstenberg and Katznelson used this structural analysis of measure-preserving systems to provide a perspicuous proof of Szemerédi’s theorem. Beleznay and Foreman showed that, in general, the transfinite construction of the maximal distal factor of a separable measure-preserving system can extend arbitrarily far into the countable ordinals. Here we show that the Furstenberg-Katznelson proof does not require the full strength of the maximal distal factor, in the sense that the proof only depends on a combinatorial weakening of its properties. We show that this combinatorially weaker property obtains fairly low in the transfinite construction, namely, by the ωωω th level. 1.