Isomorphisms and slow entropy of deterministic $[T,T^{-1}]$ systems

We study skew product measure-preserving systems driven by an irrational rotation and the step function with values $1$ and $-1$ on each half of the circle. These systems can be seen as a deterministic version of the $[T,T^{-1}]$ system, and are particular instances of Rokhlin cocycle extensions. Under the assumption of ergodicity, we show that these systems obey an interesting rigidity property, in the sense that only obvious isomorphisms are possible. We prove that two skew products are isomorphic if and only if the rotations in the base are isomorphic, and the fiber transformations are flip isomorphic. Under mild extra assumptions, we are able to fully characterize all possible isomorphism maps. We prove that by choosing the rotation angle in the base suitably, these skew products can realize arbitrarily low measure-theoretic complexity in the sense of lower slow entropy. We apply this result to obtain new examples of systems that fail the variational principle for slow entropy. These examples can have a set of invariant measures as rich as desired (any metrizable Choquet simplex, up to affine homeomorphism).