标题： 离散$μ$-二分谱：超越一致性与新见解 显示英文标题

标题： Discrete $μ$-dichotomy spectrum: beyond uniformity and new insights

摘要： 我们为非自治线性差分系统发展了谱定理，考虑了不同类型的$\mu$-二分法，包括一致和非一致的。 在非一致情况下，出现了引人入胜的情形——这些情形已被使用但其后果尚未被彻底探索——这些情形表现出意想不到的非传统行为。 这些特殊情况促使我们引入了非自治系统的两个新属性（即使在连续时间框架中也是如此），这些属性似乎在现有文献中被忽略了。 此外，我们引入了一个新的非一致$\mu$-二分法谱概念，它介于传统的非一致$\mu$-二分法谱和缓慢的非一致$\mu$-二分法谱之间。 而且，这一点尤其值得注意，我们提出了一个猜想，使得在这个新环境中可以推导出谱定理。 最后，与近年来的看法相反，通过最优比映射的视角，我们证明了弱运动相似系统之间的非一致指数二分法谱并不保持不变。 显示英文摘要

摘要： We develop spectral theorems for nonautonomous linear difference systems, considering different types of $\mu$-dichotomies, both uniform and nonuniform. In the nonuniform case, intriguing scenarios emerge -- that have been employed but whose consequences have not been thoroughly explored -- which surprisingly exhibit unconventional behavior. These particular cases motivate us to introduce two novel properties of nonautonomous systems (even in the continuous-time framework), which appear to have been overlooked in the existing literature. Additionally, we introduce a new conceptualization of a nonuniform $\mu$-dichotomy spectrum, which lies between the traditional nonuniform $\mu$-dichotomy spectrum and the slow nonuniform $\mu$-dichotomy spectrum. Moreover, and this is particularly noteworthy, we propose a conjecture that enables the derivation of spectral theorems in this new setting. Finally, contrary to what has been believed in recent years, through the lens of optimal ratio maps, we show that the nonuniform exponential dichotomy spectrum is not preserved between systems that are weakly kinematically similar.