AI 中文总结
该研究通过连续可观测量探讨拓扑动力系统的一致刚性与弱混合，明确了SIP、IP相关刚性可观测量的代数结构及因子构造条件，给出弱混合的等价判定准则。
AI 中文摘要
我们通过连续可观测量研究一致刚性与拓扑弱混合。对于固定的时间序列，沿该序列刚性的可观测量构成一个闭的单位$T^{\pm1}$不变代数，并确定沿指定序列一致刚性的极大因子。随后我们给出经典$\text{SIP}^*$与$\text{IP}^*$返回时间准则的函数形式：拓扑动力系统是弱混合的，当且仅当它无非平凡局部SIP刚性可观测量；在极小范畴中，该性质等价于无非平凡局部IP刚性可观测量。最后，局部IP刚性可观测量产生一个带有标记局部数据的典范轨道名因子。对于固定局部数据，当局部刚性代数的$T^{\pm1}$不变核非平凡时，该核确定一个一致刚性因子。
英文摘要
We study uniform rigidity and topological mild mixing through continuous observables. For a fixed sequence of times, the observables rigid along that sequence form a closed unital $T^{\pm1}$-invariant algebra and determine the maximal factor uniformly rigid along the prescribed sequence. We then give functional forms of the classical ${\rm SIP}^{*}$- and ${\rm IP}^{*}$-return-time criteria: a topological dynamical system is mildly mixing exactly when it has no nonconstant locally SIP-rigid observable, and in the minimal category the same property is equivalent to the absence of nonconstant locally IP-rigid observables. Finally, a locally IP-rigid observable yields a canonical orbit-name factor carrying marked local data. For fixed local data, the $T^{\pm 1}$-invariant core of the local rigidity algebra determines a uniformly rigid factor whenever the core is nontrivial.
