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In order to adapt to the differentiable setting a formula for linear responseproved by Pollicott and Vytnova in the analytic setting, we show a result ofregularity of dynamical determinants of expanding maps of the circle. The maintool is the decomposition of a transfer operator as a sum of a nuclear part anda "small" bounded part.

Parameter regularity of dynamical determinants of expanding maps of the circle and an application to linear response