通过亚稳定性视角看分段扩张映射的ACIM不稳定性

中文摘要 AI 辅助

受Keller的W形映射及其变体的启发，我们引入了一类一般的扩张映射族，使得扰动映射在极限映射的不动点附近具有一个收缩的几乎不变邻域。扰动映射的唯一绝对连续不变测度（ACIM）可以收敛到各种类型的极限测度。我们确定了一个局部量，它决定了极限是绝对连续的、奇异的，还是这两者的非平凡凸组合。此外，对于非平凡凸组合的情况，我们证明了扰动系统的动力学在适当的慢时间尺度下收敛到一个跳跃马尔可夫过程，这种收敛还推广到有界变差可观测量的扩散系数。与具有亚稳定行为的扩张映射的类似结果相比，我们设置的一个新特征是马尔可夫过程出现一个局部化状态，该状态对应于不动点周围收缩的几乎不变区间。我们的方法提供了一个一般框架，特别地，据我们所知，该框架容纳了所有先前研究的收敛到Keller的W形映射的族。

英文摘要

Motivated by Keller's W-shaped maps and its variants, we introduce a general class of families of expanding maps such that the perturbed maps have a shrinking almost invariant neighborhood about a fixed point of the limit map. The unique absolutely continuous invariant measures (ACIM) of the perturbed maps can converge to limit measures of various types. A local quantity is identified which determines if the limit is absolutely continuous, singular, or a non-trivial convex combination of these. Furthermore, for the case of a nontrivial convex combination, we prove that the dynamics of the perturbed system, when viewed on an appropriate slow time scale, converges to a jump Markov process, a convergence that extends to the diffusion coefficients for observables of bounded variation. Compared to analogous results on expanding maps with metastable behavior, a new feature of our setting is the emergence of a localized state of the Markov process which corresponds to the shrinking almost invariant interval about the fixed point. Our approach provides a general framework which, in particular, accommodates, to the best of our knowledge, all previously studied families that limit to Keller's W-shaped map.

发表机构

• Budapest University of Technology and Economics（布达佩斯技术与经济大学）
• HUN-REN–BME Stochastics Research Group, Budapest University of Technology and Economics（匈牙利研究网络-布达佩斯技术与经济大学随机过程研究组）

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