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Title: Sequence entropy and g-measures
Author: Hulse, Paul
ISNI:       0000 0001 3578 689X
Awarding Body: University of Warwick
Current Institution: University of Warwick
Date of Award: 1980
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This thesis presents some results on sequence entropy and g-measures. Chapter I is concerned with the sequence entropy hΑ(T) of transformations with quasi-discrete 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 quasi-discrete 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 non-zero 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 quasi-discrete spectrum but not discrete spectrum and a larger class for which hA(T) is infinite whenever T has quasi-eigen 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 In [13] and unpublished work by 'Walters, sup hA(T) was calculated for ergodic transformations.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 weak-mixing and 3trong-raixing 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 g-measures. It is not known if g-measures 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.
Supervisor: Not available Sponsor: Science Research Council
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA Mathematics