Differentiating the Absolutely Continuous Invariant Measure of an Interval Map f with Respect to f

Abstract

Let the map f:[−1,1]→[−1,1] have a.c.i.m. ρ (absolutely continuous f-invariant measure with respect to Lebesgue). Let δρ be the change of ρ corresponding to a perturbation X=δf○f⁻¹ of f. Formally we have, for differentiable A,

but this expression does not converge in general. For f real-analytic and Markovian in the sense of covering (−1,1) m times, and assuming an analytic expanding condition, we show that

is meromorphic in C, and has no pole at λ=1. We can thus formally write δρ(A)=Ψ(1).