Related papers: A dynamical condition for differentiability of Mat…

In this article we study the differentiability of Mather's $\beta$-function on closed surfaces and its relation to the integrability of the system.

We prove that if a time-periodic Tonelli Lagrangian on a closed manifold $M$ satisfies a strong version of the Differentiability Problem for Mather's $\beta$-function, then the Legendre transforms of rational homology classes are dense in…

We study Lagrangian systems on a closed manifold. We link the differentiability of Mather's beta-function with the topological complexity of the complement of the Aubry set. As a consequence, when the dimension of the manifold is less than…

Let L be a time-periodic Lagrangian on a two-torus. Then the beta-function of L is differentiable at least in k directions at any k-irrational homology class, for k= 0, 1, 2.

Using methods from symplectic topology, we prove existence of invariant variational measures associated to the flow $\phi_H$ of a Hamiltonian $H\in C^{\infty}(M)$ on a symplectic manifold $(M,\omega)$. These measures coincide with Mather…

We consider the minimal average action (Mather's $\beta$ function) for area preserving twist maps of the annulus. The regularity properties of this function share interesting relations with the dynamics of the system. We prove that the…

We investigate rigidity phenomena associated to the stable norm and Mather's $\beta$-function for Riemannian geodesic flows on closed manifolds. Given two metrics $g_1$ and $g_2$, we compare these objects pointwise at individual homology…

We review the author's results on Mather's $\beta$ function : non-strict convexity of $\beta$ when the configuration space has dimension two, link between the size of the Aubry set and the differentiability of $\beta$, correlation between…

Aubry-Mather is traditionally concerned with Tonelli Hamiltonian (convex and super-linear). In \cite{Vi,MVZ}, Mather's $\alpha$ function is recovered from the homogenization of symplectic capacities. This allows the authors to extend the…

In this paper we discuss the Mather problem for stationary Lagrangians, that is Lagrangians $L:\Rr^n\times \Rr^n\times \Omega\to \Rr$, where $\Omega$ is a compact metric space on which $\Rr^n$ acts through an action which leaves $L$…

We study properties of action-minimizing invariant sets for Tonelli Lagrangian and Hamiltonian systems and weak KAM solutions to the Hamilton-Jacobi equation in terms of Mather's averaging functions. Our principal discovery is that exposed…

Effective Hamiltonians arise in several problems, including homogenization of Hamilton--Jacobi equations, nonlinear control systems, Hamiltonian dynamics, and Aubry--Mather theory. In Aubry--Mather theory, related objects, Mather measures,…

With the variational method introduced by J Mather, we construct a mechanical Hamiltonian system whose Alpha function has a flat F and is non-differentiable at the boundary of F. In the case of two degrees of freedom, we prove this…

For a conformal vector field on a closed, real-analytic, Lorentzian manifold we prove that the flow is locally isometric -- that it preserves a metric in the conformal class on a neighborhood of any point -- or the metric is everywhere…

We present further developments on the Lagrangian 1-form description for one-dimensional integrable systems in both discrete and continuous levels. A key feature of integrability in this context called a closure relation will be derived…

We consider one-dimensional classical time-dependent Hamiltonian systems with quasi-periodic orbits. It is well-known that such systems possess an adiabatic invariant which coincides with the action variable of the Hamiltonian formalism. We…

We show that a necessary and sufficient condition for a smooth function on the tangent bundle of a manifold to be a Lagrangian density whose action can be minimized is, roughly speaking, that it be the sum of a constant, a nonnegative…

The space of deformations of the integer Heisenberg group under the action of $\textrm{Aut}(H(\mathbb{R}))$ is a homogeneous space for a non-reductive group. We analyze its structure as a measurable dynamical system and obtain mean and…

In this article we discuss a weaker version of Liouville's theorem on the integrability of Hamiltonian systems. We show that in the case of Tonelli Hamiltonians the involution hypothesis on the integrals of motion can be completely dropped…

On metric spaces equipped with doubling measures, we prove that a differentiability theorem holds for Lipschitz functions if and only if the space supports nontrivial (metric) derivations in the sense of Weaver that satisfy an additional…