Linear Response for Contracting on Average Iterated Function Systems

TL;DR

This paper investigates the differentiability of the stationary measure's integral with respect to a parameter in a probabilistic iterated function system, establishing conditions for differentiability and non-differentiability.

Contribution

It provides new results on when the integral of a test function against the stationary measure is differentiable with respect to system parameters, including generalizations.

Findings

Identifies three cases where the parameter-dependent integral is differentiable.

Shows the derivative matches the formal derivative in these cases.

Provides conditions for the existence of non-differentiable cases with smooth test functions.

Abstract

Consider the following probabilistic contracting on average iterated function system $$\Phi = \left\{f_i (x) = \lambda_i x + d_i,\;i=1,2 ;\;\; p = \left(\frac{1}{2} , \frac{1}{2}\right) \right\},$$ where the contraction ratios $\lambda_1 , \lambda_2$ are such that $0<\lambda_1<1<\lambda_2$ and $\lambda_1\lambda_2<1$. Denote by $\mu_{\lambda_1,\lambda_2}$ its stationary measure. We study the differentiability of $$(\heartsuit)\quad\quad\quad\quad\quad \lambda_1 \mapsto \int_{\mathbb{R}} \phi(x) \,d\mu_{\lambda_1,\lambda_2}(x),$$ where $\phi$ is a suitable test function. We establish three cases where $(\heartsuit)$ is differentiable and show the derivative coincides with the one obtained by taking formal derivative, which can be generalized to the case of multiple maps with different probabilities. We also present sufficient conditions under which there exists a smooth, bounded test function $\phi$ so that $(\heartsuit)$ is not differentiable.