A formula for the Entropy of the Convolution of Gibbs probabilities on the circle

Consider the transformation $T:S^1\to S^1$, such that $T(x)=2x$ (mod 1), and where $S^1$ is the unitary circle. Suppose $J:S^1\to \mathbb{R}$ is Holder continuous and positive, and moreover that, for any $y\in S^1$, we have that ${\sum }_{x\text{such that}T(x)=y}J(x)=1.$ We say that $\rho$ is a Gibbs probability for the Holder continuous potential $\log J$, if ${\mathcal{L}}_{\log J}^{\ast }(\rho )=\rho ,$ where $$\mathcal{L}$_{\log J}$istheRuelleoperatorfor$\log J$\mathrm{.}Wecall$J$theJacobianof$\rho$. Suppose $\nu ={\mu }_1\ast {\mu }_2$ is the convolution of two Gibbs probabilities ${\mu }_1$ and ${\mu }_2$ associated, respectively, to $\log J_1$ and $\log J_2$. We show that $\nu$ is also Gibbs and its Jacobian $\tilde{J}$ is given by $\tilde{J}(u)=\int J_1(u-x)d{\mu }_2(x)$ In this case, the entropy $h(\nu )$ is given by the expression $$h(\nu )=-\int [\int \log (\int J_1(r+s-x)d{\mu }_2(x))d{\mu }_2(r)]d{\mu }_1(s)\mathrm{.}$$ For a fixed ${\mu }_2$ we consider differentiable variations ${\mu }_1^t$, $t\in (-ϵ,ϵ)$, of ${\mu }_1$ on the Banach manifold of Gibbs probabilities, where ${\mu }_1^0={\mu }_1$, and we estimate the derivative of the entropy $h({\mu }_1^t\ast {\mu }_2)$ at $t=0$. We also present an expression for the Jacobian of the convolution of a Gibbs probability $\rho$ with the invariant probability with support on a periodic orbit of period two. This expression is based on the Jacobian of $\rho$ and two Radon-Nidodym derivatives.