AI 中文总结
该研究针对具有公共尖点的随机复合区间映射，在特定尖点指数下证明退火转移算子的谱隙，给出小扰动下平稳密度的存在唯一性及线性响应公式。
AI 中文摘要
我们研究具有公共尖点和公共尖点值的尖点帐篷型区间映射的独立同分布随机复合。对于尖点指数$-1<\beta<-\frac{1}{2}$，我们证明对应的退火转移算子在$W^{1,1}(I)$和$W^{2,1}(I)$上存在谱隙。对于概率律的所有足够小的扰动，存在平稳密度$h_\varepsilon\in W^{2,1}(I)$，且在属于$W^{1,1}(I)$的平稳密度中唯一。此外，映射$\varepsilon\mapsto h_\varepsilon$在$W^{1,1}(I)$中于$\varepsilon=0$处可微，我们得到了显式线性响应公式。
英文摘要
We study i.i.d.\ random compositions of cusp tent-like interval maps having a common cusp point and a common cusp value. For cusp exponents $-1<β<-\frac12$, we prove that the associated annealed transfer operators have a spectral gap on $W^{1,1}(I)$ and $W^{2,1}(I)$. For all sufficiently small perturbations of the probability law, there is a stationary density $h_\varepsilon\in W^{2,1}(I)$, unique among stationary densities belonging to $W^{1,1}(I)$. Moreover, the map $\varepsilon\mapsto h_\varepsilon$ is differentiable at $\varepsilon=0$ in $W^{1,1}(I)$, and we obtain an explicit linear response formula.
Comments26 pages, one figure
