标题： 密集的和稠密共轭类在零维动力系统中 显示英文标题

标题： Dense and comeager conjugacy classes in zero-dimensional dynamics

摘要： 设$G$是一个可数群。 我们考虑所有$G$在康托空间上通过同胚作用的波兰空间，并研究该空间中是否存在一个稠密共轭类以及一些自然子空间。 我们还开发了一个一般性的模型论框架来研究这个问题及相关问题。 我们证明，对于一个有限生成的自由群，在最小作用的空间中以及在最小的、概率测度保持作用的空间中，都存在一个稠密共轭类。 我们还确定了这两个类：第一个是所有索菲最小移位的弗赖谢极限，第二个是通用的有限阶作用。 在相反的方向上，如果$G$是一个非有限生成的可解群，我们证明在所有作用的空间中不存在稠密共轭类，如果$G$是局部有限的，则在最小作用的空间中也不存在稠密共轭类。 最后，我们研究了在拓扑传递作用空间中是否存在一个稠密共轭类的问题。 我们证明，如果$G$是自由的或虚拟幂零的，则这样的稠密共轭类存在当且仅当$G$是虚拟循环的。 显示英文摘要

摘要： Let $G$ be a countable group. We consider the Polish space of all actions of $G$ on the Cantor space by homeomorphisms and study the existence of a comeager conjugacy class in this space and some natural subspaces. We also develop a general model-theoretic framework to study this and related questions. We prove that for a finitely generated free group, there is a comeager conjugacy class in the space of minimal actions, as well as in the space of minimal, probability measure-preserving actions. We also identify the two classes: the first one is the Fra\"iss\'e limit of all sofic minimal subshifts and the second, the universal profinite action. In the opposite direction, if $G$ is an amenable group which is not finitely generated, we show that there is no comeager conjugacy class in the space of all actions and if $G$ is locally finite, also in the space of minimal actions. Finally, we study the question of existence of a dense conjugacy class in the space of topologically transitive actions. We show that if $G$ is free or virtually polycyclic, then such a dense conjugacy class exists iff $G$ is virtually cyclic.