We study the spectral properties of ergodic Schrödinger operators that are associated with a certain family of non-primitive substitutions on a binary alphabet. The corresponding subshifts provide examples of dynamical systems that go beyond minimality, unique ergodicity and linear complexity. In some parameter region, we are naturally in the setting of an infinite ergodic measure. The almost sure spectrum is singular and contains an interval. We show that under certain conditions, eigenvalues can appear. Some criteria for the exclusion of eigenvalues are fully characterized, including the existence of strongly palindromic sequences. Many of our structural insights rely on return word decompositions in the context of non-uniformly recurrent sequences. We introduce an associated induced system that is conjugate to an odometer.

我们研究了遍历薛定谔算子的谱特性，这些算子与二进制字母表上的某个非原始替换族相关。相应的子位移提供了超越极小性、独特遍历性和线性复杂性的动力系统示例。在某些参数区域，我们自然处于无限遍历测度的设置中。几乎可以肯定的频谱是奇异的并且包含一个区间。我们表明，在某些条件下，可以出现特征值。一些排除特征值的标准是完全表征的，包括强回文序列的存在。我们的许多结构见解依赖于非均匀循环序列上下文中的返回词分解。我们引入了一个与里程表共轭的相关诱导系统。