We provide an ergodic theory framework to study statistical properties of smooth sequences over the odd alphabet {1,3}. The arithmetic nature of this alphabet yields a partition of the subshift of smooth sequences based on their local structure, defining a notion of type for those sequences. We describe the substitutive structure of the smaller subshifts obtained by fixing the sequence of types of the successive derivatives of smooth sequences, from which we obtain the unique ergodicity of all these subshifts. A direct consequence is that the asymptotic frequency of any finite pattern in a smooth sequence over {1,3} is always well-defined and depends on its type sequence. Finally, we characterize the minimality of these subshifts.

翻译：我们提供了一个遍历理论框架，用于研究奇数字母表 $\{1,3\}$ 上光滑序列的统计性质。该字母表的算术性质，根据局部结构对光滑序列的子移位进行了划分，从而定义了这些序列的类型概念。我们描述了通过固定光滑序列逐次导数的类型序列所获得的更小子移位的代换结构，并由此证明了所有这些子移位的唯一遍历性。直接结论是，$\{1,3\}$ 上光滑序列中任何有限模式的渐近频率总是明确定义的，且依赖于其类型序列。最后，我们刻画了这些子移位的最小性。