Differentiating the absolutely continuous invariant measure of an interval map f with respect to f

Abstract

Let the map $f:[-1,1]\to [-1,1]$ have a.c.i.m. $\rho$ (absolutely continuous $f$-invariant measure with respect to Lebesgue). Let $\delta \rho$ be the change of $\rho$ corresponding to a perturbation $X=\delta f\circ f^{-1}$ of $f$. Formally we have, for differentiable $A$, $$\delta \rho (A)=\sum _{n=0}^{\infty }\int \rho (dx)X(x)\frac{d}{dx}A(f^{n}x)$$ 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 {\it analytic expanding} condition, we show that $$\lambda \mapsto \Psi (\lambda )=\sum _{n=0}^{\infty }{\lambda }^n\int \rho (dx)X(x)\frac{d}{dx}A(f^{n}x)$$ is meromorphic in $\mathbf{C}$, and has no pole at $\lambda =1$. We can thus formally write $\delta \rho (A)=\Psi (1)$.

10 pages, plain TeX

References in corpus (1)

• Differentiation of SRB states for hyperbolic flows

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