Isomorphisms and slow entropy of deterministic [T,T−1] systems
Abstract
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 Kalikow's [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 for isomorphisms. That is, we show 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 characterize all isomorphisms. We prove that by choosing the rotation angle suitably, these systems 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).
Related
- Isomorphisms of ℓ1-crossed products of topological systems
- Slow Entropy of Quasi-Unipotent Actions on S-Arithmetic Homogeneous Spaces
- STRONG INVARIANCE PRINCIPLES IN DETERMINISTIC SYSTEMS
- The entry-exit function and nilpotent corrections in multidimensional slow-fast systems
- Curvature-Dependent Path Concentration in Stochastic Fast-Slow Systems with Noise on the Slow Variable
BibTeX
@misc{carrascovargas2026isomorphisms,
title = {{Isomorphisms and slow entropy of deterministic $[T,T^-1]$ systems}},
author = {Nicanor Carrasco-Vargas},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13589},
eprint = {2609.13589},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13589},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Isomorphisms and slow entropy of deterministic [T,T^−1] systems
AU - Nicanor Carrasco-Vargas
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.13589
UR - https://arxiv.org/abs/2609.13589
AB - 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 Kalikow's [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 for isomorphisms. That is, we show 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 characterize all isomorphisms. We prove that by choosing the rotation angle suitably, these systems 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).
ER -