Linear Response Of Piecewise Expanding Interval Maps

In this paper.we mainly study the differentiability of absolutely continuous invariant measures induced by piecewise expanding interval maps,give a sufficient condition for its differentiability.and derive the corresponding linear response formula.By analyzing the transfer operator in the space of functions with bounded variation.we can obtain the linear response formula of the absolutely continuous invariant measure.The main difficulty of the study is that the expression is possibly not convergent by the influence of singularities.We prove that the linear response formula is valid if the field of perturbation is zero on all image sets of the singular set.