Abstract
In this paper we give explicit characterizations, based on the cutting and spacer parameters, of (a) which rank-one transformations factor onto a given finite cyclic permutation, (b) which rank-one transformations factor onto a given odometer, and (c) which rank-one transformations are isomorphic to a given odometer. These naturally yield characterizations of (d) which rank-one transformations factor onto some (unspecified) finite cyclic permutation, (d′) which rank-one transformations are totally ergodic, (e) which rank-one transformations factor onto some (unspecified) odometer, and (f) which rank-one transformations are isomorphic to some (unspecified) odometer.
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Acknowledgment
The research in this paper was done at the AIM SQuaRE titled The isomorphism problem for rank-one transformations. The authors would like to acknowledge the American Institute of Mathematics for the support on this research. M.F. acknowledges the US NSF grants DMS-1700143 and DMS-2100367 for support for this research. S.G. acknowledges the US NSF grants DMS-1201290 and DMS-1800323 for the support of his research. From August 2019 to August 2021, C.S. served as a Program Director in the Division of Mathematical Sciences at the National Science Foundation (NSF), USA, and as a component of this job, he received support from NSF for research, which included work on this paper. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. We would like to thank the referee for a careful reading and suggestions that shortened our proofs.
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Foreman, M., Gao, S., Hill, A. et al. Rank-one transformations, odometers, and finite factors. Isr. J. Math. 255, 231–249 (2023). https://doi.org/10.1007/s11856-022-2451-y
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DOI: https://doi.org/10.1007/s11856-022-2451-y