Y. Katznelson’s research while affiliated with Stanford University and other places

LetX be a probability space and letf: X n → {0, 1} be a measurable map. Define the influence of thek-th variable onf, denoted byI f (k), as follows: Foru=(u 1,u 2,…,u n−1) ∈X n−1 consider the setl k (u)={(u 1,u 2,...,u k−1,t,u k ,…,u n−1):t ∈X}. If (k) = Pr(u Î Xn - 1 :f is not constant on lk (u)).I_f (k) = \Pr (u \in X^{n - 1} :f is not constant on l_k (u)). More generally, forS a subset of [n]={1,...,n} let the influence ofS onf, denoted byI f (S), be the probability that assigning values to the variables not inS at random, the value off is undetermined. Theorem 1:There is an absolute constant c 1 so that for every function f: X n → {0, 1},with Pr(f −1(1))=p≤1/2,there is a variable k so that If (k) \geqslant c1 p\fraclognn.I_f (k) \geqslant c_1 p\frac{{\log n}}{n}. Theorem 2:For every f: X n → {0, 1},with Prob(f=1)=1/2, and every ε>0,there is S ⊂ [n], |S|=c 2(ε)n/logn so that I f (S)≥1−ε. These extend previous results by Kahn, Kalai and Linial for Boolean functions, i.e., the caseX={0, 1}.

Zero entropy processes are known to be deterministic—the past determines the present. We show that each is isomorphic, as a system, to a finitarily deterministic one, i.e., one in which to determine the present from the past it suffices to scan a finite (of random length) portion of the past. In fact we show more: the finitary scanning can be done even if the scanner is noisy and passes only a small fraction of the readings, provided the noise is independent of our system. The main application we present here is that any zero entropy system can be extended to a random Markov process (namely one in which the conditional distribution of the present given the past is a mixture of finite state Markov chains). This allows one to study zero entropy transformations using a procedure completely different from the usual cutting and stacking.

A simpler treatment is given of the theorems of Dye and Krieger concerning the classification of non-singular ergodic transformations up to orbit equivalence.

We shall present here two examples from “geometric Ramsey theory” which illustrate how ergodic theoretic techniques can be used to prove that subsets of Euclidean space of positive density necessarily contain certain configurations. Specifically we will deal with subsets of the plane, and our results will be valid for subsets of “positive upper density”.

Our purpose in this paper is to present a more or less complete solution to the problem of the smoothness of the conjugation of aperiodic diffeomorphisms of the circle. We show that the rotation number and the smoothness of the diffeomorphism guarantee a certain smoothness for the homeomorphism which conjugates it with a rigid rotation, and obtain the best smoothness that can be guaranteed.

Let f be an orientation preserving -diffeomorphism of the circle. If the rotation number α = ρ(f) is irrational and log Df is of bounded variation then, by a wellknown theorem of Denjoy, f is conjugate to the rigid rotation Rα. The conjugation means that there exists an essentially unique homeomorphism h of the circle such that f = h−lRαh. The general problem of relating the smoothness of h to that of f under suitable diophantine conditions on α has been studied extensively (cf. [H1], [KO], [Y] and the references given there). At the bottom of the scale of smoothness for f there is a theorem of M. Herman [H2] which states that if Df is absolutely continuous and D log Df Lp, p > 1, α = ρ (f) is of ‘constant type’ which means ‘the coefficients in the continued fraction expansion of α are bounded’, and if f is a perturbation of Rα, then h is absolutely continuous. Our purpose in this paper is to give a different proof and an improved version of Herman's theorem. The main difference in the result is that we do not need to assume that f is close to Rα; the proof is very different from Herman's and is very much in the spirit of [KO].(Received June 22 1987)(Revised August 27 1988)

We prove a theorem about idempotents in compact semigroups. This theorem gives a new proof of van der Waerden’s theorem on arithmetic progressions as well as the Hales-Jewett theorem. It also gives an infinitary version of the Hales-Jewett theorem which includes results of T. J. Carlson and S. G. Simpson.

This chapter discusses the density version of the Hales–Jewett theorem for k = 3 and outlines the main elements of the proof. The method used is “ergodic” and the first step is to show that the theorem in question can be formulated as a statement regarding a certain family of measure preserving transformations acting on a (probability) measure space. The phenomenon that appears in the treatment of Szemeredi's theorem, is that when a family of measure preserving transformations having a certain structure acts on a measure space, a set of positive measure will necessarily return to itself (“recur”) under certain combinations of transformations. This recurrence phenomenon for sets of positive measure is then translated into the appearance of certain patterns in subsets of a sufficiently large structure, provided the density of the subset is bounded.