In this paper, we associate a primitive substitution with a family of non-integer positional number systems with respect to the same base but with different sets of digits. In this way, we generalize the classical Dumont–Thomas numeration which corresponds to one specific case. Therefore, our concept also covers beta-expansions induced by Parry numbers. But we establish links to variants of beta-expansions such as symmetric beta-expansions, too. In other words, we unify several well-known notions of non-integer representations within one general framework. A focus in our research is set on finiteness and periodicity properties. It turns out that these characteristics mainly depend on the substitution. As a consequence we are able to relate known finiteness properties that are viewed independently yet.

中文翻译：

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代换数制

在本文中，我们将原始替换与一系列非整数位置数字系统相关联，这些系统具有相同的基数但具有不同的数字集。通过这种方式，我们概括了对应于一个特定情况的经典 Dumont-Thomas 计数。因此，我们的概念还涵盖了由 Parry 数引起的 beta 扩展。但我们也建立了与 beta 扩展变体的链接，例如对称 beta 扩展。换句话说，我们在一个通用框架内统一了几个众所周知的非整数表示概念。我们研究的重点是有限性和周期性属性。事实证明，这些特征主要取决于替代。因此，我们能够关联尚未独立查看的已知有限性属性。