Relative slow entropy

In 1997, Katok–Thouvenot and Ferenczi independently introduced a notion of ``slow entropy’’ as a way to quantitatively compare measure-preserving systems with zero entropy. We develop a relative version of this theory for a measure-preserving system conditioned on a given factor. Our new 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. As an application, we prove a relative version of a result of Ferenczi that classifies isometric systems in terms of their slow entropy. We also introduce a new definition for the notion of a rigid extension and investigate its relationship to relative slow entropy.