退火Ruelle-Pollicott共振
Annealed Ruelle-Pollicott Resonances
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中文总结 AI 辅助
该研究将Ruelle-Pollicott共振理论推广至退火随机动力系统，定义共振并建立去相关公式，通过区间分段扩张马尔可夫映射及逆谱问题明确了相关理论的应用。
中文摘要 AI 辅助
我们将Ruelle-Pollicott共振理论推广到由独立同分布映射族生成的退火随机动力系统中。引入退火转移算子与Koopman算子，我们将共振定义为相关算子点谱的元素，并建立了将这些共振与退火关联的渐近衰减关联起来的去相关公式。随后，我们研究了几类可明确应用该理论的系统：首先考虑区间上的分段扩张马尔可夫映射族，构造适配动力学的巴拿赫空间，得到退火共振谱的完整描述；接着研究逆谱问题，证明了给定复数集合可作为适当构造的退火动力系统的共振的可实现性结果。
英文摘要
We adapt the theory of Ruelle-Pollicott resonances to annealed random dynamical systems generated by independent and identically distributed families of maps. Introducing annealed transfer and Koopman operators, we define resonances as elements of the point spectrum of the associated operators and establish a decorrelation formula relating these resonances to the asymptotic decay of annealed correlations. We then study several classes of systems for which the theory can be made explicit. First, we consider a family of piecewise expanding Markov maps of the interval, we construct Banach spaces adapted to the dynamics and obtain a complete description of the annealed resonance spectrum. Then we investigate an inverse spectral problem, proving realisability results for prescribed collections of complex numbers as resonances of suitably constructed annealed dynamical systems.
