For a contracting-on-average IFS with affine maps where one contracts and one expands but their product is less than one, the expectation of suitable test functions under the stationary measure is differentiable in three cases with the derivative matching the formal one, while non-differentiable for
Random walks in the group of Eu- clidean isometries and self-similar measures
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Linear Response for Contracting on Average Iterated Function Systems
For a contracting-on-average IFS with affine maps where one contracts and one expands but their product is less than one, the expectation of suitable test functions under the stationary measure is differentiable in three cases with the derivative matching the formal one, while non-differentiable for