On free minimal constant speedups violating continuous orbit equivalence in p-adic ℤd-odometers of adding type
Abstract
Let ℤp be the ring of p-adic integers with respect to a prime p and let d be a positive integer. For each z=(z1, z2,..., zd) ∈ ℤpd, let Tz: ℤd × ℤp → ℤp be an adding-type ℤd-action on ℤp defined by Tnz(x):=x+ ∑i=1d ni zi for n=(n1,n2, ⋯, nd) ∈ ℤd and x ∈ ℤp. Under some mild assumptions on z, the action Tz is a free ℤd-odometer (by odometer, we mean a minimal and equicontinuous action on a Cantor space). In this paper, we derive a necessary condition for continuous orbit equivalence between such ℤd-odometers by constructing algebraic models for them. We then study the free minimal constant speedups of these ℤd-odometers. It turns out that such a speedup of Tz is again an adding-type p-adic ℤd-odometer Tw for some w ∈ ℤpd. However, the necessary condition above may not hold for the speedup. This provides the first known examples of free minimal bounded speedups (of free ℤd-odometers) which are not continuously orbit equivalent to the original ones and hence disproves a conjecture by Johnson and McClendon. Our result also indicates that continuous orbit equivalence is a rare phenomenon for free minimal constant speedups of p-adic ℤd-odometers of adding type when d geqslant 2.
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BibTeX
@misc{jiao2026free,
title = {{On free minimal constant speedups violating continuous orbit equivalence in $p$-adic $\mathbbZ^d$-odometers of adding type}},
author = {Changhua Jiao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01450},
eprint = {2609.01450},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01450},
}
RIS
TY - UNPB
TI - On free minimal constant speedups violating continuous orbit equivalence in p-adic ℤ^d-odometers of adding type
AU - Changhua Jiao
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.01450
UR - https://arxiv.org/abs/2609.01450
AB - Let ℤ_p be the ring of p-adic integers with respect to a prime p and let d be a positive integer. For each z=(z_1, z_2,..., z_d) ∈ ℤ_p^d, let T_z: ℤ^d × ℤ_p → ℤ_p be an adding-type ℤ^d-action on ℤ_p defined by T^n_z(x):=x+ ∑_i=1^d n_i z_i for n=(n_1,n_2, ⋯, n_d) ∈ ℤ^d and x ∈ ℤ_p. Under some mild assumptions on z, the action T_z is a free ℤ^d-odometer (by odometer, we mean a minimal and equicontinuous action on a Cantor space). In this paper, we derive a necessary condition for continuous orbit equivalence between such ℤ^d-odometers by constructing algebraic models for them. We then study the free minimal constant speedups of these ℤ^d-odometers. It turns out that such a speedup of T_z is again an adding-type p-adic ℤ^d-odometer T_w for some w ∈ ℤ_p^d. However, the necessary condition above may not hold for the speedup. This provides the first known examples of free minimal bounded speedups (of free ℤ^d-odometers) which are not continuously orbit equivalent to the original ones and hence disproves a conjecture by Johnson and McClendon. Our result also indicates that continuous orbit equivalence is a rare phenomenon for free minimal constant speedups of p-adic ℤ^d-odometers of adding type when d geqslant 2.
ER -