On free minimal constant speedups violating continuous orbit equivalence in $p$-adic $\mathbb{Z}^d$-odometers of adding type

Abstract: Let $\mathbb{Z}<em>p$ be the ring of $p$-adic integers with respect to a prime $p$ and let $d$ be a positive integer. For each $$\mathbf{z}=(z_1, z_2,..., z_d) \in \mathbb{Z}_p<sup>d$$, let $$T</em>{\mathbf{z}}: \mathbb{Z}^d \times \mathbb{Z}<em>p \to \mathbb{Z}_p$$ be an adding-type $\mathbb{Z}<sup>d$-action on $\mathbb{Z}_p$ defined by $$T^{\mathbf{n}}</em>{\mathbf{z}}(x):=x+ \sum_{i=1}^d n_i z_i$$ for $$\mathbf{n}=(n_1,n_2, \cdots, n_d) \in \mathbb{Z}<sup>d$$ and $x \in \mathbb{Z}<em>p$. Under some mild assumptions on $\mathbf{z}$, the action $T</em>{\mathbf{z}}$ is a free $\mathbb{Z}<sup>d$-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 $\mathbb{Z}<sup>d$-odometers by constructing algebraic models for them. We then study the free minimal constant speedups of these $\mathbb{Z}<sup>d$-odometers. It turns out that such a speedup of $T_{\mathbf{z}}$ is again an adding-type $p$-adic $\mathbb{Z}<sup>d$-odometer $T_{\mathbf{w}}$ for some $\mathbf{w} \in \mathbb{Z}_p<sup>d$. However, the necessary condition above may not hold for the speedup. This provides the first known examples of free minimal bounded speedups (of free $\mathbb{Z}<sup>d$-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 $\mathbb{Z}<sup>d$-odometers of adding type when $d \geqslant 2$.