On the Susceptibility Function of Piecewise Expanding Interval Maps
On the Susceptibility Function of Piecewise Expanding Interval Maps
复制标题
分段展开区间图的磁敏函数
DOI：
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发表时间：
2006
期刊：
影响因子：
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通讯作者：
V. Baladi
中科院分区：
文献类型：
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作者：
V. Baladi
摘要研究了磁化率函数
$$Psi(z)=sum_{n=0}^infty int z^n X (y)
ho_0(y) frac {partial}{partial y} varphi (f^n (y)), dy$$与分段展开区间映射f的扰动$$f_t,=,f,+,tX,circ,f$$有关，并与一个可观测值φ有关。Ψ(1)是φ的平均值$${mathcal{R}}(t),=,intvarphi
ho_t,dx$$相对于ft的SRB度量的形式导数（在t = 0时）。我们的分析是基于传递算子的谱描述。给出了f、X和φ上的充分条件，保证Ψ(z)在大于1的圆盘上是全纯的，或保证一个数可以与可能的发散级数Ψ(1)相关联。我们给出了f， X和φ的例子，使得$${mathcal{R}}(t)$$在0处不是Lipschitz，并且我们提出了Ruelle猜想的一个新版本。
AbstractWe study the susceptibility function
$$Psi(z)=sum_{n=0}^infty int z^n X (y)
ho_0(y) frac {partial}{partial y} varphi (f^n (y)), dy$$associated to the perturbation $$f_t,=,f,+,tX,circ,f$$ of a piecewise expanding interval map f, and to an observable φ. Ψ(1) is the formal derivative (at t = 0) of the average $${mathcal{R}}(t),=,intvarphi
ho_t,dx$$ of φ with respect to the SRB measure of ft. Our analysis is based on a spectral description of transfer operators. It gives in particular sufficient conditions on f, X, and φ which guarantee that Ψ(z) is holomorphic in a disc of larger than one, or which ensure that a number may be associated to the possibly divergent series Ψ(1) . We present examples of f, X, and φ so that $${mathcal{R}}(t)$$ is not Lipschitz at 0, and we propose a new version of Ruelle’s conjecture.