Title:

Sequence entropy and gmeasures

This thesis presents some results on sequence entropy and gmeasures. Chapter I is concerned with the sequence entropy hΑ(T) of transformations with quasidiscrete spectrum. In [10], it was shown that T has discrete spectrum if and only if hA(T) is zero for all sequences A. This prompts the question: If T has quasidiscrete spectrum but not discrete spectrum, for which sequences is hΑ(T) positive? Y/e first consider this problem for affine transformations on the torus and calculate the sequence entropies for certain types of sequence. In the general case, we obtain sufficient conditions on the sequence for zero and nonzero sequence entropy. With a suitable restriction imposed on the sequence, we get a necessary and sufficient condition for zero sequence entropy. Next, we determine a class of sequences for which HA(T) is infinite whenever T has quasidiscrete spectrum but not discrete spectrum and a larger class for which hA(T) is infinite whenever T has quasieigen functions of arbitrarily large order. An example is given to show that this last result does not characterize such transformations. In [13] and unpublished work by 'Walters, sup hA(T) was calculated for ergodic transformations. In Chapter II, §1, using the same method, we extend this result to show that the supremum is always attained. We then deduce necessary and sufficient conditions for weakmixing and 3trongraixing in terms of sequence entropy, strengthening similar results in [15]. In §2, we use a construction in [5] to construct sets W of arbitrarily small measure such that {TniW}⧟1 generates the full ⧜algebra, where { Tni}⧟1 “ converges weakly to the identity. By combining this with the results of $1, we deduce the existence of a transformation T and a sequence A such that hA(T) is infinite and there exist subsequence generators for A with arbitrarily small entropy. This contrasts with the case A = {n}, where the existence of a generator with finite entropy implies the entropy of T i3 finite and is the infimum of the entropies of the generators. In Chapter III, we consider the uniqueness problem for gmeasures. It is not known if gmeasures are unique in general. However, a sufficient condition for uniqueness in terms of the variation of log g has been given in [18], We construct examples to show this condition is not necessary for uniqueness.
