洛伦兹型映射的线性响应公式

Linear Response Formula of Lorenz-type Maps

洛伦兹型映射是具有不连续点的分段扩张映射,其不连续性来源于展示蝴蝶效应的洛伦兹方程的奇点,该映射可观测的统计性质由绝对连续的不变测度给出.本文考虑一类改进的洛伦兹型映射f的扰动f_(t)=f+tXof,对应绝对连续测度μ的扰动记为μ_(t).我们证明如果X在f的不连续点集的所有像集上取值为零,则它的敏感性公式Ψ(λ)=∞Σn=0λ∫μ(dx)X(x)■(φ(f^(n)x))/■x,φ∈C(1),在λ=1处是收敛的,从而得到线性响应公式d/dt|t=0μ_(t)(φ)=Ψ(1)成立.

The Lorenz-type maps are piecewise expanding maps with discontinuous points,the discontinuity comes from the singularities of Lorenz equations showing butterfly effect,and the observable statistical properties of such maps are given by the absolutely continuous invariant measures.In this paper,we consider the perturbation f_(t)=f+tXof of an improved Lorenz-type map f,and denote byμ_(t)the perturbation of the corresponding absolutely continuous measureμ.We prove that if X takes zero on all image sets of the discontinuous point of f,then its sensitivity formulaΨ(λ)=∞Σn=0λ∫μ(dx)X(x)■(φ(f^(n)x))/■x,φ∈C(1),converges atλ=1,thus the linear response formula d/dt|t=0μ_(t)(φ)=Ψ(1)is established.