Linear Response for Contracting on Average Iterated Function Systems

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.