A Borel ${\mathbb{Z}}^d$ dynamical system $(X,S)$ is ‘almost Borel universal’ if any free Borel ${\mathbb{Z}}^d$ dynamical system $(Y,T)$ of strictly lower entropy is isomorphic to a Borel subsystem of $(X,S)$, after removing a null set. We obtain and exploit a new sufficient condition for a topological ${\mathbb{Z}}^d$ dynamical system to be almost Borel universal. We use our main result to deduce various conclusions and answer a number of questions. Along with additional results, we prove that a ‘generic’ homeomorphism of a compact manifold of topological dimension at least two can model any ergodic transformation, that non-uniform specification implies almost Borel universality, and that 3-colorings in ${\mathbb{Z}}^d$ and dimers in ${\mathbb{Z}}^2$ are almost Borel universal.

中文翻译：

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Borel 子系统和通过规范及超越的紧凑 Zd 系统的遍历普遍性

一个波雷尔 ${\mathbb{Z}}^d$ 动力系统 $(X,秒)$ 如果有任何免费的 Borel，则“几乎是 Borel 通用的” ${\mathbb{Z}}^d$ 动力系统 $(是,吨)$ 严格低熵的同构于一个 Borel 子系统 $(X,秒)$, 删除空集后。我们获得并利用拓扑的新充分条件${\mathbb{Z}}^d$动力系统几乎是Borel通用的。我们使用我们的主要结果来推导出各种结论并回答一些问题。连同额外的结果，我们证明了拓扑维数至少为 2 的紧凑流形的“通用”同胚可以模拟任何遍历变换，非均匀规范意味着几乎 Borel 普遍性，并且 3-colorings in${\mathbb{Z}}^d$ 和二聚体 ${\mathbb{Z}}^2$ 几乎是 Borel 通用的。