论文标题
关于某些分段双曲图中线性响应公式的收敛
On convergence of linear response formulae in some piecewise hyperbolic maps
论文作者
论文摘要
当高维非均匀的双曲混沌系统发生动态扰动时，通常会观察到它们的长期统计数据对扰动有所不同。尽管在应用中很重要，但这种可不同的性能被认为与系统的维度有关，它仍然对严格的研究具有抵抗力。为了对非均匀双曲线系统进行建模，我们考虑了一个数学上可拖延的一类平滑双曲线图的家族，即Lozi地图。对于这些地图，我们证明，当在地图的奇异性集中进行条件时，响应的形式衍生物的存在减少了SRB度量的指数混合特性。该属性似乎是真实的，并且具有独立的利益。对这种条件混合特性的进一步研究可能会产生线性响应理论的更好情况。
When high-dimensional non-uniformly hyperbolic chaotic systems undergo dynamical perturbations, their long-time statistics are generally observed to respond differentiably with respect to the perturbation. Although important in applications, this differentiability, which is thought to be connected to the dimensionality of the system, has remained resistant to rigorous study. To model non-uniformly hyperbolic systems, we consider a family of the mathematically tractable class of piecewise smooth hyperbolic maps, the Lozi maps. For these maps we prove that the existence of a formal derivative of the response reduces to an exponential mixing property of the SRB measure when conditioned on the map's singularity set. This property appears to be true and is of independent interest. Further study of this conditional mixing property may yield a better picture of linear response theory.
