发表机构
IPME RAS（俄罗斯科学院控制问题研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究固定动力系统下加权相体积的演化，通过加权散度给出均匀指数收缩与膨胀的充分条件，揭示耗散性，并刻画全局吸引子，同时提供微分几何解释及对偶性。加权体积衰减被视为积分性质，而非逐点稳定性。
AI 中文摘要
研究了固定动力系统下加权相体积的演化，采用任意实数指数的正权重。获得了传输加权体积均匀指数收缩和膨胀的充分条件，这些条件以相应的加权散度表示。这些条件使得揭示普通散度可能无法检测到的耗散性质成为可能。建立了对不变集的推论，并在紧吸收集的假设下，刻画了全局吸引子的加权测度。该方法通过距离加权环境体积估计扩展到不变子流形。还提供了微分几何解释，其中加权散度与共形变换度量相关的普通散度等同。对指数的依赖用于揭示互逆权重之间的对偶性。此外，利用变号权重导出了不变超曲面指数吸引的充分局部条件。加权体积衰减被解释为传输集的积分性质，其本身并不等同于逐点李雅普诺夫稳定性。
英文摘要
The evolution of weighted phase volume under a fixed dynamical system is investigated using a positive weight raised to an arbitrary real exponent. Sufficient conditions for uniform exponential contraction and expansion of transported weighted volume are obtained in terms of the corresponding weighted divergence. These conditions make it possible to reveal dissipative properties that may remain undetected by the ordinary divergence. Consequences for invariant sets are established, and, under the assumption of a compact absorbing set, the weighted measure of a global attractor is characterized. The approach is extended to invariant submanifolds through a distance-weighted ambient-volume estimate. A differential-geometric interpretation is also provided, in which the weighted divergence is identified with the ordinary divergence associated with a conformally transformed metric. The dependence on the exponent is used to reveal a duality between mutually inverse weights. In addition, sufficient local conditions for exponential attraction to an invariant hypersurface are derived using a sign-changing weight. Weighted-volume decay is interpreted as an integral property of transported sets and, by itself, is not identified with pointwise Lyapunov stability.
