To every dynamical system $(X,\varphi)$ over a totally disconnected compact space, we associate a left-orderable group $T(\varphi)$. It is defined as a group of homeomorphisms of the suspension of $(X,\varphi)$ which preserve every orbit of the suspension flow and act by dyadic piecewise linear homeomorphisms in the flow direction. We show that if the system is minimal, the group is simple, and if it is a subshift then the group is finitely generated. The proofs of these two statements are short and elementary, providing straightforward examples of finitely generated simple left-orderable groups. We show that if the system is minimal, every action of the corresponding group on the circle has a fixed point. These constitute the first examples of finitely generated left-orderable groups with this fixed point property. We show that for every system $(X,\varphi)$, the group $T(\varphi)$ does not have infinite subgroups with Kazhdan's property $(T)$. In addition, we show that for every minimal subshift, the corresponding group is never finitely presentable. Finally if $(X,\varphi)$ has a dense orbit then the isomorphism type of the group $T(\varphi)$ is a complete invariant of flow equivalence of the pair $\{\varphi, \varphi^{-1}\}$. In the appendix we describe a Polish group into which $T(\varphi)$ embeds densely.

中文翻译：

---

流的分段线性同胚群

对于完全断开的紧空间上的每个动力系统 $(X,\varphi)$，我们关联一个左序群 $T(\varphi)$。它被定义为 $(X,\varphi)$ 的悬浮的一组同胚，它们保留悬浮流的每个轨道，并在流动方向上通过二元分段线性同胚起作用。我们证明，如果系统是最小的，则群是简单的，如果是子位移，则群是有限生成的。这两个陈述的证明简短而基本，提供了有限生成的简单左序群的直接例子。我们证明，如果系统是最小的，则相应组在圆上的每个动作都有一个固定点。这些构成了具有这种不动点特性的有限生成左序群的第一个例子。我们证明对于每个系统 $(X,\varphi)$，群 $T(\varphi)$ 没有具有 Kazhdan 性质 $(T)$ 的无限子群。此外，我们表明，对于每个最小子位移，相应的组永远不会是有限可表示的。最后，如果 $(X,\varphi)$ 有一个稠密轨道，则群 $T(\varphi)$ 的同构类型是对 $\{\varphi, \varphi^{-1 的流等价的完全不变量}\}$。在附录中，我们描述了 $T(\varphi)$ 密集嵌入的波兰语群。\varphi)$ 有一个稠密轨道，则群 $T(\varphi)$ 的同构类型是 $\{\varphi, \varphi^{-1}\}$ 对的流动等价的完全不变量。在附录中，我们描述了 $T(\varphi)$ 密集嵌入的波兰语群。\varphi)$ 有一个稠密轨道，则群 $T(\varphi)$ 的同构类型是 $\{\varphi, \varphi^{-1}\}$ 对的流动等价的完全不变量。在附录中，我们描述了 $T(\varphi)$ 密集嵌入的波兰语群。