具有嵌套不变锥的系统的统计行为
Statistical behavior of systems with nested invariant cones
- University of Alberta（阿尔伯塔大学）
- Nanjing University of Science and Technology（南京理工大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究嵌套不变锥约束下动力系统的统计行为，证明Birkhoff中心和不变测度支集的连通分量位于单一锥层，并在横截条件下可嵌入低维空间，应用于反馈系统和抛物方程得到零拓扑熵。
AI中文摘要：
嵌套不变锥（NICs）在许多有限维和无限维动力系统中编码了层次序和振荡结构。我们研究了这种层次结构如何约束Birkhoff中心以及不变概率测度的支集。对于相对于NICs一致最终强单调的最终紧致、耗散半流，我们证明了Birkhoff中心的每个连通分量$B$位于单个锥层中：存在$j>0$使得$B$相对于所有较低层锥是无序的，并且相对于第$j$层或更高层的所有锥是强有序的。同样的结论对于不变概率测度的支集的每个连通分量也成立。在涉及余维为$d$的线性子空间的额外横截性条件下，每个这样的分量允许嵌入到$\mathbb R^d$中。如果$d=1$或$2$，并且半流在其全局吸引子上的限制扩展为流，则系统具有零拓扑熵，与原始相空间的维数无关。我们将该理论应用于双向循环反馈系统和圆上的标量抛物方程。在抛物情形中，自然的零数NICs的无限族即使对于热方程也不满足一致最终强单调性；我们通过构造有限族扰动NICs来克服这一障碍。在这两个应用中，Birkhoff中心的每个连通分量都允许平面嵌入，并且拓扑熵为零。
英文摘要：
Nested invariant cones (NICs) encode hierarchical order and oscillation structures in many finite and infinite dimensional dynamical systems. We investigate how this hierarchy constrains the Birkhoff center and the supports of invariant probability measures. For eventually compact, dissipative semiflows that are uniformly eventually strongly monotone with respect to NICs, we prove that every connected component $B$ of the Birkhoff center lies in a single cone layer: there exists $j>0$ such that $B$ is unordered with respect to all lower-level cones and strongly ordered with respect to all cones at level $j$ or higher. The same conclusion holds for every connected component of the support of an invariant probability measure. Under an additional transversality condition involving a codimension-$d$ linear subspace, each such component admits a homeomorphic embedding into $\mathbb R^d$. If $d=1$ or $2$ and the restriction of the semiflow to its global attractor extends to a flow, then the system has zero topological entropy, independently of the dimension of the original phase space. We apply the theory to bidirectional cyclic feedback systems and scalar parabolic equations on the circle. In the parabolic case, the natural infinite family of zero-number NICs fails to be uniformly eventually strongly monotone, even for the heat equation; we overcome this obstruction by constructing a finite family of perturbed NICs. In both applications, every connected component of the Birkhoff center admits a planar embedding, and the topological entropy is zero.
