AI 中文总结
本文研究加法型p进ℤ^d-odometer的自由极小常加速，发现其与原odometer可能不连续轨道等价，否定了Johnson和McClendon的猜想，指出d≥2时这类加速的连续轨道等价现象罕见。
AI 中文摘要
设ℤ_p为关于素数p的p进整数环，d为正整数。对每个z=(z₁,z₂,…,z_d)∈ℤ_p^d，定义加法型ℤ^d-作用T_z:ℤ^d×ℤ_p→ℤ_p，满足对任意n=(n₁,n₂,…,n_d)∈ℤ^d、x∈ℤ_p，有T_z^n(x)=x+∑_{i=1}^d n_i z_i。在z满足某些温和假设时，作用T_z是自由ℤ^d-odometer（odometer指Cantor空间上的极小等连续作用）。本文通过为这类ℤ^d-odometer构造代数模型，推导了它们之间连续轨道等价的必要条件，随后研究了这类ℤ^d-odometer的自由极小常加速。结果表明，T_z的这类加速仍是某个w∈ℤ_p^d对应的加法型p进ℤ^d-odometer T_w，但上述必要条件对加速未必成立。这提供了首批自由ℤ^d-odometer的自由极小有界加速例子，这类加速与原odometer不连续轨道等价，从而否定了Johnson和McClendon的猜想；还表明当d≥2时，连续轨道等价对加法型p进ℤ^d-odometer的自由极小常加速而言是罕见现象。
英文摘要
Let $\mathbb{Z}_p$ be the ring of $p$-adic integers with respect to a prime $p$ and let $d$ be a positive integer. For each $\mathbf{z}=(z_1, z_2,..., z_d) \in \mathbb{Z}_p^d$, let $T_{\mathbf{z}}: \mathbb{Z}^d \times \mathbb{Z}_p \to \mathbb{Z}_p$ be an adding-type $\mathbb{Z}^d$-action on $\mathbb{Z}_p$ defined by $T^{\mathbf{n}}_{\mathbf{z}}(x):=x+ \sum_{i=1}^d n_i z_i$ for $\mathbf{n}=(n_1,n_2, \cdots, n_d) \in \mathbb{Z}^d$ and $x \in \mathbb{Z}_p$. Under some mild assumptions on $\mathbf{z}$, the action $T_{\mathbf{z}}$ is a free $\mathbb{Z}^d$-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 $\mathbb{Z}^d$-odometers by constructing algebraic models for them. We then study the free minimal constant speedups of these $\mathbb{Z}^d$-odometers. It turns out that such a speedup of $T_{\mathbf{z}}$ is again an adding-type $p$-adic $\mathbb{Z}^d$-odometer $T_{\mathbf{w}}$ for some $\mathbf{w} \in \mathbb{Z}_p^d$. However, the necessary condition above may not hold for the speedup. This provides the first known examples of free minimal bounded speedups (of free $\mathbb{Z}^d$-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 $\mathbb{Z}^d$-odometers of adding type when $d \geqslant 2$.
