Department of Mathematics, Rutgers University, Piscataway, NJ 08854, USA.
Philos Trans A Math Phys Eng Sci. 2011 Jan 28;369(1935):482-93. doi: 10.1098/rsta.2010.0260.
Let ρ be a Sinai-Ruelle-Bowen (SRB or 'physical') measure for the discrete time evolution given by a map f, and let ρ(A) denote the expectation value of a smooth function A. If f depends on a parameter, the derivative δρ(A) of ρ(A) with respect to the parameter is formally given by the value of the so-called susceptibility function Ψ(z) at z=1. When f is a uniformly hyperbolic diffeomorphism, it has been proved that the power series Ψ(z) has a radius of convergence r(Ψ)>1, and that δρ(A)=Ψ(1), but it is known that r(Ψ)<1 in some other cases. One reason why f may fail to be uniformly hyperbolic is if there are tangencies between the stable and unstable manifolds for (f,ρ). The present paper gives a crude, non-rigorous, analysis of this situation in terms of the Hausdorff dimension d of ρ in the stable direction. We find that the tangencies produce singularities of Ψ(z) for |z|<1 if d<1/2, but only for |z|>1 if d>1/2. In particular, if d>1/2, we may hope that Ψ(1) makes sense, and the derivative δρ(A)=Ψ(1) thus has a chance to be defined.
设 ρ 为给定离散时间演化的西奈-鲁埃尔-鲍温（SRB 或“物理”）测度，由映射 f 给出，并且设 ρ(A) 表示光滑函数 A 的期望价值。如果 f 取决于参数，则 ρ(A) 相对于参数的导数 δρ(A)形式上由所谓的易感染函数 Ψ(z)在 z=1 处的值给出。当 f 是一致双曲微分同胚时，已经证明了幂级数 Ψ(z)具有收敛半径 r(Ψ)>1，并且 δρ(A)=Ψ(1)，但已知在其他一些情况下 r(Ψ)<1。f 可能不是一致双曲的原因之一是稳定流形和不稳定流形之间可能存在相切。本文以稳定方向上 ρ 的豪斯多夫维数 d 为指标，对这种情况进行了粗略的、非严格的分析。我们发现，如果 d<1/2，则相切会在 |z|<1 时产生 Ψ(z)的奇点，但如果 d>1/2，则仅在 |z|>1 时产生奇点。特别地，如果 d>1/2，我们可能希望 Ψ(1)有意义，并且导数 δρ(A)=Ψ(1)因此有机会被定义。
