One property of trajectories of Toeplitz flows

基于托普利茨（Toeplitz）矩阵的具有泛用性质的深度Koopman分层模型

Deep Koopman-layered Model with Universal Property Based on Toeplitz Matrices

本研究提出基于Toeplitz矩阵的深度Koopman分层模型，兼具理论性与灵活性，可拟合非自治系统时序数据，结合Krylov子空间方法高效训练，在Koopman算子特征值估计任务上性能优于现有方法。

Toeplitz flows and their ordered K-theory

To a Toeplitz flow $(X,T)$ we associate an ordered $K^0$-group, denoted $K^0(X,T)$, which is order isomorphic to the $K^0$-group of the associated (non-commutative) $C^\ast$-crossed product $C(X)\rtimes_T \mathbb{Z}$. However, $K^0(X,T)$ can be defined in purely dynamical terms, and it turns out to be a complete invariant for (strong) orbit equivalence. We characterize the $K^0$-groups that arise from Toeplitz flows $(X,T)$ as exactly those simple dimension groups $(G,G^+)$ that contain a noncyclic subgroup $H$ of rank one that intersects $G^+$ nontrivially. Furthermore, the Bratteli diagram r…

Rigidity of joinings for some measure preserving systems

We introduce two properties: strong R-property and $C(q)$-property, describing a special way of divergence of nearby trajectories for an abstract measure preserving system. We show that systems satisfying the strong R-property are disjoint (in the sense of Furstenberg) with systems satisfying the $C(q)$-property. Moreover, we show that if $u_t$ is a unipotent flow on $G/Γ$ with $Γ$ irreducible, then $u_t$ satisfies the $C(q)$-property provided that $u_t$ is not of the form $h_t\times\operatorname{id}$, where $h_t$ is the classical horocycle flow. Finally, we show that the strong R-property hol…