Consider the transformation , such that (mod 1), and where S1 is the unitary circle. Suppose is Hölder continuous and positive, and moreover that, for any , we have that We say that ρ is a Gibbs probability for the Hölder continuous potential , if where is the Ruelle operator for . We call J the Jacobian of ρ.Suppose is the convolution of two Gibbs probabilities and associated, respectively, to and . We show that ν is also Gibbs and its Jacobian is given by .In this case, the entropy is given by the expressionFor a fixed we consider differentiable variations , , of on the Banach manifold of Gibbs probabilities, where , and we estimate the derivative of the entropy at t = 0.We also present an expression for the Jacobian of the convolution of a Gibbs probability ρ with the invariant probability with support on a periodic orbit of period two. This expression is based on the Jacobian of ρ and two Radon–Nidodym derivatives.