  In Nonequilibrium Thermodynamics and Information Theory, the relative entropy
(or, KL divergence) plays a very important role. Consider a H\"older Jacobian
$J$ and the Ruelle (transfer) operator $\mathcal{L}_{\log J}.$ Two equilibrium
probabilities $\mu_1$ and $\mu_2$, can interact via a discrete-time {\it
Thermodynamic Operation} described by the action {\it of the dual of the Ruelle
operator} $ \mathcal{L}_{\log J}^*$. We argue that the law $\mu \to
\mathcal{L}_{\log J}^*(\mu)$, producing nonequilibrium, can be seen as a
Thermodynamic Operation after showing that it's a manifestation of the Second
Law of Thermodynamics. We also show that the change of relative entropy
satisfies $$ D_{K L} (\mu_1,\mu_2) - D_{K L} (\mathcal{L}_{\log
J}^*(\mu_1),\mathcal{L}_{\log J}^*(\mu_2))= 0.$$ Furthermore, we describe
sufficient conditions on $J,\mu_1$ for getting $h(\mathcal{L}_{\log
J}^*(\mu_1))\geq h(\mu_1)$, where $h$ is entropy. Recalling a natural
Riemannian metric in the Banach manifold of H\"older equilibrium probabilities
we exhibit the second-order Taylor formula for an infinitesimal tangent change
of KL divergence; a crucial estimate in Information Geometry. We introduce
concepts like heat, work, volume, pressure, and internal energy, which play
here the role of the analogous ones in Thermodynamics of gases. We briefly
describe the MaxEnt method.
