Given a positive integer M and $q \in (1, M+1]$ we consider expansions in base q for real numbers $x \in [0, {M}/{q-1}]$ over the alphabet $\{0, \ldots , M\}$ . In particular, we study some dynamical properties of the natural occurring subshift $(\boldsymbol{{V}}_q, \sigma )$ related to unique expansions in such base q. We characterize the set of $q \in \mathcal {V} \subset (1,M+1]$ such that $(\boldsymbol{{V}}_q, \sigma )$ has the specification property and the set of $q \in \mathcal {V}$ such that $(\boldsymbol{{V}}_q, \sigma )$ is a synchronized subshift. Such properties are studied by analysing the combinatorial and dynamical properties of the quasi-greedy expansion of q. We also calculate the size of such classes as subsets of $\mathcal {V}$ giving similar results to those shown by Blanchard [ 10 ] and Schmeling in [ 36 ] in the context of $\beta $ -transformations.

中文翻译：

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关于唯一q-展开的规范性质和同步

给定一个正整数米和$q \in (1, M+1]$我们考虑扩大基地q对于实数$x \in [0, {M}/{q-1}]$在字母表上$\{0, \ldots , M\}$. 特别是，我们研究了自然发生的子位移的一些动力学特性$(\boldsymbol{{V}}_q, \sigma )$与此类基地的独特扩展有关q. 我们表征集合$q \in \mathcal {V} \subset (1,M+1]$这样$(\boldsymbol{{V}}_q, \sigma )$有规范属性和集合$q \in \mathcal {V}$这样$(\boldsymbol{{V}}_q, \sigma )$是一个同步的子移位。这些性质是通过分析准贪心展开的组合性质和动力学性质来研究的q. 我们还将这些类的大小计算为$\数学{V}$给出了与 Blanchard [ 10 ] 和 Schmeling 在 [ 36 ] 中在$\beta $-转换。