无轨道唯一性的动力系统的遍历性
Ergodicity of dynamical systems without uniqueness of orbits
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中文总结 AI 辅助
本文针对非确定性动力系统，提出基于强后向不变性的遍历性定义，并证明经典结果及Birkhoff遍历定理的类似结论成立。
中文摘要 AI 辅助
近年来，非确定性动力系统的研究引起了相当大的兴趣。为了从测度论的角度分析此类系统的混沌行为，考虑遍历性是必要的。然而，遍历性的经典定义涉及不变集，而对于非确定性动力系统，不变集的定义并不唯一。因此，我们面临这样一个问题：哪种不变性能够产生有趣的遍历性定义。在此，我们提出一种基于强后向不变性的定义，并证明经典结果的类似结论成立。我们还考虑了Birkhoff遍历定理对无轨道唯一性系统的影响。
英文摘要
Recently, there has been considerable interest in the study of non-deterministic dynamical systems. To analyze the chaotic behavior of such systems from a measure-theoretic viewpoint, it is desirable to consider ergodicity. However, the classical definition of ergodicity involves invariant sets, whose definition is not unique for non-deterministic dynamical systems. Thus, we are led to the question of which invariance yields an interesting definition of ergodicity. Here, we propose a definition based on the strong backward invariance and show that analogs of classical results hold. We also consider implications of the Birkhoff ergodic theorem for systems without uniqueness of orbits.
发表机构
- Tokyo University of Science（东京科学大学）
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