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

Abstract

Consider the transformation $T:S^1 \to S^1$, such that $T(x)=2\, x$ (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 $\mathcal{L}_{\log J}^* \,(\rho)=\rho ,$ where $\mathcal{L}_{\log J}$ is the Ruelle operator for $\log J$. We call $J$ the Jacobian of $\rho$. Suppose $\nu=\mu_1*\mu_2$ 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 $$ h(\nu) = - \int\,[\,\,\int\, \log \,(\,\int J_1(r+s-x) d \mu_2(x)\,) \, d \mu_2(r)\,\, ]\,\,d \mu_1 (s).$$ For a fixed $\mu_2$ we consider differentiable variations $\mu_1^t$, $t \in (-\epsilon,\epsilon)$, of $\mu_1$ on the Banach manifold of Gibbs probabilities, where $\mu_1^0=\mu_1$, and we estimate the derivative of the entropy $h(\mu_1^t * \mu_2)$ 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.