Stanley Eigen’s research while affiliated with Northeastern University and other places

This article is concerned with demonstrating the power and simplicity of sww (special weakly wandering) sequences. We calculate an sww growth sequence for the infinite measure preserving random walk transformation. From this we obtain the first explicit eww (exhaustive weakly wandering) sequence for the transformation. The exhaustive property of the eww sequence is a “gift” from the sww sequence and requires no additional work. Indeed we know of no other method for finding explicit eww sequences for the random walk map or any other infinite ergodic transformation. The result follows from a detailed analysis of the proof of Theorem 3.3.12 in the book S.Eigen, A.Hajian, Y.Ito, V.Prasad, Weakly Wandering Sequences in Ergodic Theory (Springer, Tokyo, 2014) as applied to the random walk transformation from which an sww growth sequence is obtained.We explain the significance of sww sequences in the construction of eww sequences.

The appearance of weakly wandering (ww) sets and sequences for ergodic transformations over half a century ago was an unexpected and surprising event. In time it was shown that ww and related sequences reflected significant and deep properties of ergodic transformations that preserve an infinite measure. This monograph studies in a systematic way the role of ww and related sequences in this classification of ergodic transformations preserving an infinite measure. Unexpected to connections of these sequences to additive number theory (in particular tilings of the integers) are also discussed. As the tools for finite ergodic theory are inadequate for the development in this book, no knowledge of ergodic theory is presumed. We assume only a basic knowledge of Lebesgue measure theory.

In this chapter we study infinite tilings of the integers and we explore the connection between infinite tilings of the integers and infinite ergodic theory. This will reveal a structure for certain eww sequences. With this structure we will extend the examples of Chap. 4.

In this chapter we discuss properties of infinite sequences of integers associated with an infinite ergodic transformation. We note that all the sequences we consider are isomorphism invariants for such transformations.

In this chapter we present and discuss three basic examples of infinite ergodic transformations. We show some special and unique properties these transformations possess. These properties involve characteristics of infinite ergodic transformations that distinguish the transformations from finite ergodic transformations.

In this chapter we examine several isomorphism invariants associated to an infinite ergodic transformation. We use these invariants to show the non-isomorphism of various infinite ergodic transformations. In Sect. 6.1 the role of eww sets and sequences is discussed and used to distinguish between some infinite ergodic transformations. In Sect. 6.2 it is shown that there exists a class of infinite ergodic transformations that exhibit a regularity in the size of the return sets of finite measure. We define (an isomorphism invariant) the α-type for such transformations. Previously, in Sect. 3.3 recurrent sequences for an infinite ergodic transformation were defined, and then in Sect. 4.1 the First Basic Example was constructed and all its recurrent sequences were computed. This is extended in Sect. 6.3 to a family of transformations where all the recurrent sequences for the members of that family are computed and shown to distinguish between any two members of the family. Finally, in Sect. 6.4 the growth distribution of ww sets is also shown to be an effective isomorphism invariant.

In the previous chapters we saw that recurrent transformations do not accept wandering sets. An important subset of the recurrent transformations are the ergodic ones that do not have a finite invariant and equivalent measure. These transformations also do not accept wandering sets, yet they must necessarily accept ww and eww sets. For infinite ergodic transformations the existence of ww sets is a significant property and reflects the subtle features of these transformations. In this chapter we discuss the strong bond that exists between infinite ergodic transformations and ww or eww sequences.

About fifty years ago, questions on the existence and non-existence of finite invariant measures were studied by various authors and from different directions. In this article, we examine these classical results and prove directly that all the conditions introduced by these authors are equivalent to each other. We begin at the fundamental level of a recurrent transformation whose properties can be strengthened to obtain the aforementioned classical results for the existence of a finite invariant measure. We conclude with the introduction of a new property, Strongly Weakly Wandering (sww) sequences, the existence of which is equivalent to the non-existence of a finite invariant measure. It is shown that every sww sequence is also an Exhaustive Weakly Wandering (eww) sequence for ergodic transformations. Although all ergodic transformations with no finite invariant measure are known to have eww sequences, there are exceedingly few actual examples for which explicit eww sequences can be exhibited. The significance of sww sequences is that it provides a condition which is easier to verify than the condition for eww sequences (Proposition 4.5). In a second paper, we will continue these studies and also connect them to some of the more recent derived conditions for finite invariant measures. The impetus for this work, began with the late Professor Shizuo Kakutani, with whom the authors worked and had many fruitful discussions on these topics.

A set of integers A is said to tile ℤ if there exists another set of integers B such that for every integer n there exist unique integers a∈A and b∈B with n=a+b. In this paper, we show the existence of an infinite tiling set, A⊂ℤ, with the hereditary property that every infinite subset A ' ⊂A also tiles ℤ. In particular, we show that every sequence which increases “fast enough” is a tiling set.