单纯形动力学中的全测历史行为与涌现
Full-measure historic behavior and emergence in simplex dynamics
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中文总结 AI 辅助
本文提出一种有限维边界驻留机制，使单纯形映射的非固定轨道产生非收敛历史行为，且历史集具全测度，并给出经验测度累积集及多项式涌现阶。
中文摘要 AI 辅助
我们给出了Takens型历史行为的有限维边界驻留机制。对于一类单纯形上的连续自映射，朝向边界的乘法Lyapunov漂移，结合在分离的边界区域附近的循环驻留，迫使每个非固定内部轨道具有非收敛的Cesàro平均值和非收敛的经验测度。因此，历史集具有全相对Lebesgue测度。该机制独立于双曲规范、游荡域和正熵：事实上，对于这里考虑的映射，拓扑熵完全由边界限制承载。非收敛性在所有正幂次下持续存在，更一般地，在乘法厚采样时间下也持续存在。抽象准则适用于Stein--Ulam和Lotka--Volterra型随机算子，一旦验证了模型特定的循环编码和对数驻留估计。在显式循环环模型中，我们确定了经验测度的完整弱-*累积集；该累积集是一个非平凡的概率测度圆，并产生阶为\\(\varepsilon^{-1}\\)的多项式逐点涌现。我们还通过将该机制与Barański--Misiurewicz型边界追踪构造相结合，获得了具有正维omega极限集的例子。
英文摘要
We give a finite-dimensional boundary-residence mechanism for Takens-type historic behavior. For a class of continuous self-maps of a simplex, a multiplicative Lyapunov drift toward the boundary, combined with cyclic residence near separated boundary regions, forces every non-fixed interior orbit to have non-convergent Cesàro averages and non-convergent empirical measures. Thus the historic set has full relative Lebesgue measure. The mechanism is independent of hyperbolic specification, wandering domains and positive entropy: in fact, for the maps considered here the topological entropy is carried entirely by the boundary restriction. The non-convergence persists under all positive powers and, more generally, under multiplicatively thick sampling times. The abstract criterion applies to Stein--Ulam and Lotka--Volterra type stochastic operators once the model-specific cyclic coding and logarithmic residence estimates are verified. In explicit cyclic-collar models we identify the full weak-* accumulation set of empirical measures; this accumulation set is a non-trivial circle of probability measures and yields polynomial pointwise emergence of order \(\varepsilon^{-1}\). We also obtain examples with positive-dimensional omega-limit sets by combining the mechanism with a Barański--Misiurewicz type boundary-tracing construction.
发表机构
- United Arab Emirates University（阿联酋大学）
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