Given a finite-to-one map T acting on a compact metric space Ω and an appropriate Banach space of functions X (Ω), one classically constructs for each potential A ∈ X a transfer operator L A acting on X (Ω). Under suitable hypotheses, it is well-known that L A has a maximal eigenvalue λ A , has a spectral gap and defines a unique Gibbs measure µ A . Moreover there is a unique normalized potential of the form B := A + f − f • T + c acting as a representative of the class of all potentials defining the same Gibbs measure.The goal of the present article is to study the geometry of the set of normalized potentials N , of the normalization map A → B, and of the Gibbs map A → µ A . We give an easy proof of the fact that N is an analytic submanifold of X and that the normalization map is analytic; we compute the derivative of the Gibbs map; last we endow N with a natural weak Riemannian metric (derived from the asymptotic variance) with respect to which we compute the gradient flow induced by the pressure with respect to a given potential, e.g. the metric entropy functional. We also apply these ideas to recover in a wide setting existence and uniqueness of equilibrium states, possibly under constraints.

We prove a sharp, dimension-free stability result for the classical logarithmic Sobolev inequality for a two parameter family of functions. Roughly speaking, our family consists of a certain class of log C 1,1 functions. Moreover, we show how to enlarge this space at the expense of the dimensionless constant and the sharp exponent. As an application we obtain new bounds on the entropy.

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We discuss first-order stationary mean-field games (MFG) on networks. These models arise in traffic and pedestrian flows. First, we address the mathematical formulation of first-order MFG on networks, including junction conditions for the Hamilton-Jacobi (HJ) equation and transmission conditions for the transport equation. Then, using the current method, we convert the MFG into a system of algebraic equations and inequalities. For critical congestion models, we show how to solve this system by linear programming.

We study the Lagrangian structure of relativistic Vlasov systems, such as the relativistic Vlasov-Poisson and the relativistic quasi-eletrostatic limit of Vlasov-Maxwell equations. We show that renormalized solutions of these systems are Lagrangian and that these notions of solution, in fact, coincide. As a consequence, finite-energy solutions are shown to be transported by a global flow. Moreover, we extend the notion of generalized solution for "effective" densities, and we prove the existence of such solutions. Finally, under a higher integrability assumption of the initial condition, we show that solutions have every energy bounded, even in the gravitational case. These results extend to our setting those recently obtained for the Vlasov-Poisson system in a series of papers by Ambrosio, Colombo, and Figalli; here, we analyze relativistic systems and also consider the contribution of the magnetic force into the evolution equation.