具有收缩纤维和类Lorenz映射系统的谱间隙与定量统计稳定性
Spectral gap and quantitative statistical stability for systems with contracting fibers and Lorenz-like maps
AI总结：
研究保持收缩叶理且商映射满足Lasota-Yorke不等式的系统，证明转移算子有谱间隙并给出定量估计；对类Lorenz二维映射，证明其统计稳定性，扰动下物理测度连续模为O(δlogδ)，渐近最优。
摘要：
我们考虑保持收缩叶理（contracting foliation）的变换，使得相应的商映射满足Lasota-Yorke不等式。我们证明了作用在适当赋范空间上的相关转移算子具有谱间隙（对此我们给出了定量估计）。作为应用，我们考虑类Lorenz二维映射（具有无界收缩率和扩张率的逐片双曲映射）：我们证明这些系统具有谱间隙，并给出其统计稳定性的定量估计。在大小为$δ$的系统确定性扰动下，物理测度连续变化，其连续模为$O(δ\log δ)$，这对于这类逐片光滑映射是渐近最优的。
英文摘要：
We consider transformations preserving a contracting foliation, such that the associated quotient map satisfies a Lasota-Yorke inequality. We prove that the associated transfer operator, acting on suitable normed spaces, has a spectral gap (on which we have quantitative estimation). As an application we consider Lorenz-like two dimensional maps (piecewise hyperbolic with unbounded contraction and expansion rate): we prove that those systems have a spectral gap and we show a quantitative estimate for their statistical stability. Under deterministic perturbations of the system of size $δ$, the physical measure varies continuously, with a modulus of continuity $O(δ\log δ)$, which is asymptotically optimal for this kind of piecewise smooth maps.
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