On the Isomorphism Problem of p-Endomorphisms

On the Isomorphism Problem of p-Endomorphismsendomorphism if T is s-to-1 a.e. and the conditional probabilities of the preimages are precisely the components of p. are isomorphic if there exist a measure-preserving bijective map ϕ : X → Y such that ϕT = Sϕ a.e. This thesis considers the isomorphism problem of p-endomorphisms, generalizing the work of Hoffman and Rudolph [H,R] which treats the case when p is a uniform probability vector, i.e. p = (1/p,. .. , 1/p). In particular, we generalize the tvwB criterion introduced in Hoffman and Rudolph to prove two results. The first result is Theorem 2.4.1, which generalizes the main theorem in [H,R] to p-endomorphisms. We paraphrase this as follows: We give two proofs of this result. The first follows Ornstein's classical proof of his famous theorem that two shifts of equal entropy are isomorphic, and a second proof which follows the joinings proof as given in [H,R]. As a corollary of the joinings proof, we show that there are uncountably many automorphisms of the one-sided Bernoulli shift B + (p) unless the components of p are pairwise distinct. We also give examples of tvwB p-endomorphisms such as mixing one-sided Markov shifts and a generalization of the [T, Id] transformation. The second main result is Theorem 5.1.1, which in view of Theorem 2.4.1, reduces to the statement that for any two tvwB finite group extensions of one-sided Bernoulli shifts, there is an isomorphism between them in a stronger sense than that asserted in Theorem 2.4.1. Specifically, we have the following theorem in Chapter 5 which we paraphrase as follows: Theorem 5.1.1 ′. Let G be a finite group. For any two tvwB G-extensions of the one-sided shift B + (p), there is an isomorphism which preserves the Bernoulli factor algebra and maps fibres over points in the factor to other such fibres by group rotations. ii Acknowledgements First and foremost, I would like to thank my thesis advisor, Professor Andrés del Junco, for his insights, advice and encouragement throughout the program. Without his assistance and reassurance (especially when I have self-doubts), this thesis would not have been possible. I also wish to thank Professor Dan Rudolph for some helpful conversations during conferences in Memphis and Toronto, and for his groundbreaking work with Christopher Hoffman on which this thesis is based. I also appreciate the numerous helpful suggestions from Professors who also served on my final examination …