UCLA

Originally developed in the 1950s, Kolmogorov--Sinai entropy is a very powerful isomorphism invariant for measure-preserving systems that measures the ``complexity'' or ``randomness'' of a system. There is also a relative version of this notion which measures the additional complexity of a system when compared to a fixed reference system. In 1997, Katok--Thouvenot and Ferenczi independently introduced a notion of ``slow entropy'' as a way to quantitatively compare measure-preserving systems with zero Kolmogorov--Sinai entropy. The goal of this thesis is to develop a relative version of this theory and apply it to several other natural dynamical questions.

Chapters 1 and 2 lay out the definition and basic properties of relative slow entropy. Our definition inherits many desirable properties that make it a natural generalization of both the Katok--Thouvenot/Ferenczi theory and the classical conditional Kolmogorov--Sinai entropy. In Chapter 3, we address the question: under what conditions is a generic extension of a system also isomorphic to the base system? Using relative slow entropy as a tool to prove systems non-isomorphic, we show that a generic extension is not isomorphic to the base system whenever the base has zero Kolmogorov--Sinai entropy. Chapter 4 concerns the well-studied notions of isometric and weakly mixing extensions. We give necessary and sufficient entropy-theoretic conditions for extensions to be isometric or weakly mixing. Finally, Chapter 5 investigates the notion of rigidity. Although there is a well known definition for what it means for a single system to be rigid, there is no standard definition for the notion of a rigid extension. We provide a new candidate definition and investigate its consequences. We show that rigid extensions are generic and give an entropy-theoretic sufficient condition for an extension to be rigid. As a consequence, we obtain a new entropy-theoretic characterization of rigid systems which may be of independent interest.