Subsequence generators for ergodic group translations

Sequence Entropy and Subsequence Generators

“…It is a well-known result in ergodic theory that if T has a (two-sided) generator t, with finite entropy, then h(T) is finite and is equal to h(T, £). Moreover, h(T) is the infimum of the entropies of all generators of T. Ellis and Friedman [3] have shown that if T is ergodic and has discrete spectrum, this result holds for sequence entropy and subsequence generators. However, the following result shows that this is not the case in general.…”

Sweeping out under a collection of transformations

“…Here we shall describe an extension of the main result in [3]. In [3] it is shown that if X is a compact abelian group and 5 p contains infinitely many integral powers of a single ergodic measure-preserving rotation of X, then {A: A~ and {A, X-A} is an 5<generator} is dense in ~.…”

“…In [3] it is shown that if X is a compact abelian group and 5 p contains infinitely many integral powers of a single ergodic measure-preserving rotation of X, then {A: A~ and {A, X-A} is an 5<generator} is dense in ~. The proof grows out of two facts: the fact that a sequence of rotations can be chosen from 5 e which converges to some fixed rotation, and the fact that arbitrarily small sets can be found which sweep out X under the action of 5C By generalizing some of the Lemmas in [3], this result can be extended as follows. In light of Theorem 1 and Corollary 2, Theorem 5 shows the existence of two-set generators for many collections of transformations, in particular, for any freely acting collection containing a "Cauchy sequence".…”

“…Answering this problem involves the group structure of the group of transformations generated by ~ and the position of 5 p in this group, as well as qualities of the transformations themselves, such as their entropy and whether they admit invariant measures. We refer the reader to [1,3,4,5,6,7,8,10] for some of the work done on this problem.…”

Subset Generators for Nonsingular Transformations

“…Since countable S-generators exist for all infinite S c Z, one can ask when do there exist finite 5-generators for T, or two set S-generators for T. They do not always exist: if 7"is ergodic and measure preserving and has non-zero entropy then there are no finite N + -generators for T, and if Thas infinite entropy there are no finite generators for T. Krieger showed [14] that if T is ergodic and measure preserving then the existence and geometry of finite Z-generators for T is basically determined by the entropy of T; Krieger's result has been strengthened and extended to non-ergodic measurepreserving T in ST (see [1], [8]). For general S, however, the existence of finite S-generators for T is not well understood even for ergodic measure-preserving T. One result which is known [4] is that if T is an ergodic measure-preserving translation on a compact abelian group, then for every infinite S c Z there is a two-set S-generator for T.…”