The critical dimension as an invariant of Type III odometers

Abstract

Metric entropy is a good invariant for a useful class of measure preserving dynamical systems. This is due to metric entropy's computability and invariance under isomorphism. Many have tried to generalise metric entropy to the larger class of dynamical systems that are null-measure preserving. The problem with these proposed definitions is that they are difficult to compute. In this thesis we take one such entropy, the critical dimension, and show that with certain assumptions it is preserved under the induced transformation. This has far reaching consequences as many transformations between null-measure preserving dynamical systems are induced transformations. Hence many familiar transformations preserve the critical dimension. This allows us to compute the critical dimension for a larger range of dynamical systems, including some ITPFI factors of bounded type.