Tonelli Hamiltonians Research Articles

Diffusing orbits along chains of cylinders

<p style='text-indent:20px;'>We develop a geometric mechanism to prove the existence of orbits that drift along a prescribed sequence of cylinders, under some general conditions on the dynamics. This mechanism can be used to prove the existence of Arnold diffusion for large families of perturbations of Tonelli Hamiltonians on <inline-formula><tex-math id="M1">\begin{document}$ {{\mathbb A}}^3 $\end{document}</tex-math></inline-formula>. Our approach can also be applied to more general Hamiltonians that are not necessarily convex.</p><p style='text-indent:20px;'>The main geometric objects in our framework are <inline-formula><tex-math id="M2">\begin{document}$ 3 $\end{document}</tex-math></inline-formula>–dimensional invariant cylinders with boundary (not necessarily hyperbolic), which are assumed to admit center-stable and center-unstable manifolds. These enable us to define <i>chains of cylinders</i>, i.e., finite, ordered families of cylinders where each cylinder admits homoclinic connections, and any two consecutive cylinders in the chain admit heteroclinic connections.</p><p style='text-indent:20px;'>Our main result is on the existence of diffusing orbits which drift along such chains of cylinders, under precise conditions on the dynamics on the cylinders – i.e., the existence of Poincaré sections with the return maps satisfying a <i>tilt condition</i> – and on the geometric properties of the intersections of the center-stable and center-unstable manifolds of the cylinders – i.e., certain compatibility conditions between the <i>tilt map</i> and the homoclinic maps associated to its essential invariant circles.</p><p style='text-indent:20px;'>We give two proofs of our result, a very short and abstract one, and a more constructive one, aimed at possible applications to concrete systems.</p>

Global Generalized Characteristics for the Dirichlet Problem for Hamilton--Jacobi Equations at a Supercritical Energy Level

We study the nonhomogeneous Dirichlet problem for first-order Hamilton--Jacobi equations associated with Tonelli Hamiltonians on a bounded domain $\Omega$ of $\mathbb{R}^n$ assuming the energy leve...

Rigidity in topology C0 of the Poisson bracket for Tonelli Hamiltonians

We prove the following rigidity result for the Tonelli Hamiltonians.Let T*M be the cotangent bundle of a closed manifold M endowed with its usual symplectic form. Let (Fn) be a sequence of Tonelli Hamiltonians that C0 converges on the compact subsets to a Tonelli Hamiltonian F. Let (Gn) be a sequence of Hamiltonians that that C0 converges on the compact subsets to a Hamiltonian G. We assume that the sequence of the Poisson brackets ({Fn, Gn}) C0-converges on the compact subsets to a C1 function . Then .

Tonelli Hamiltonians without conjugate points and $$C^0$$ C 0 integrability

We prove that all the Tonelli Hamiltonians defined on the cotangent bundle $T^*\T^n$ of the $n$-dimensional torus that have no conjugate points are $C^0$ integrable, i.e. $T^*\T^n$ is $C^0$ foliated by a family $\Fc$ of invariant $C^0$ Lagrangian graphs. Assuming that the Hamiltonian is $C^\infty$, we prove that there exists a $G_\delta$ subset $\Gc$ of $\Fc$ such that the dynamics restricted to every element of $\Gc$ is strictly ergodic. Moreover, we prove that the Lyapunov exponents of every $C^0$ integrable Tonelli Hamiltonian are zero and deduce that the metric and topological entropies vanish.

On commuting Tonelli Hamiltonians: Time-periodic case

We show that the Aubry sets, the Ma\~{n}\'{e} sets and Mather's barrier functions are the same for two commuting time-periodic Tonelli Hamiltonians.

A dynamical condition for differentiability of Mather's average action

We prove the differentiability of Mather's average action on all rotation vectors of measures whose supports are contained in a Lipschitz Lagrangian asymptotically isolated graph, invariant by Tonelli Hamiltonians. We also show the relationship between differentiability of $\beta $ and local integrability of the Hamiltonian flow.

Lagrangian Rabinowitz Floer homology and twisted cotangent bundles

We study the following rigidity problem in symplectic geometry: can one displace a Lagrangian submanifold from a hypersurface? We relate this to the Arnold Chord Conjecture, and introduce a refined question about the existence of relative leaf-wise intersection points, which are the Lagrangian-theoretic analogue of the notion of leaf-wise intersection points defined by Moser (Acta. Math. 141(1–2):17–34, 1978). Our tool is Lagrangian Rabinowitz Floer homology, which we define first for Liouville domains and exact Lagrangian submanifolds with Legendrian boundary. We then extend this to the ‘virtually contact’ setting. By means of an Abbondandolo–Schwarz short exact sequence we compute the Lagrangian Rabinowitz Floer homology of certain regular level sets of Tonelli Hamiltonians of sufficiently high energy in twisted cotangent bundles, where the Lagrangians are conormal bundles. We deduce that in this situation a generic Hamiltonian diffeomorphism has infinitely many relative leaf-wise intersection points.

Weak Liouville-Arnol′d Theorems and Their Implications

This paper studies the existence of invariant smooth Lagrangian graphs for Tonelli Hamiltonian systems with symmetries. In particular, we consider Tonelli Hamiltonians with n independent but not necessarily involutive constants of motion and obtain two theorems reminiscent of the Liouville-Arnold theorem. Moreover, we also obtain results on the structure of the configuration spaces of such systems that are reminiscent of results on the configuration space of completely integrable Tonelli Hamiltonians.

Green bundles, Lyapunov exponents and regularity along the supports of the minimizing measures

In this article, we study the minimizing measures of the Tonelli Hamiltonians. More precisely, we study the relationships between the so-called Green bundles and various notions as:•the Lyapunov exponents of minimizing measures;•the weak KAM solutions. In particular, we deduce that the support of every minimizing measure μ, all of whose Lyapunov exponents are zero, is C1-regular μ-almost everywhere.

Connecting orbits for families of Tonelli Hamiltonians

We investigate the existence of Arnold diffusion-type orbits forsystems obtained by iterating in any order the time-one maps of afamily of Tonelli Hamiltonians. Such systems are known as'polysystems' or 'iterated function systems'. When specialized tofamilies of twist maps on the cylinder, our results are similar tothose obtained by Moeckel [20] and Le Calvez [15]. Ourapproach is based on weak KAM theory and is close to the one used byBernard in [3] to study the case of a single TonelliHamiltonian.

Generic Properties of Closed Orbits of Hamiltonian Flows from Mañé's Viewpoint

We show the genericity from the viewpoint of of generic properties (as symplectic linear maps) of the differential of Poincare maps of periodic orbits of Hamiltonians. Combining this result and the work of Oliveira we get Kupka-Smale type theorem a la Mane for regular energy levels of Tonelli Hamiltonians in compact manifolds. Our proof relies on techniques from geometric control theory.

On the integrability of Tonelli Hamiltonians

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 and still interesting information on the dynamics of the system can be deduced. Moreover, we prove that on the n n -dimensional torus this weaker condition implies classical integrability in the sense of Liouville. The main idea of the proof consists in relating the existence of independent integrals of motion of a Tonelli Hamiltonian to the “size” of its Mather and Aubry sets. As a byproduct we point out the existence of “non-trivial” common invariant sets for all Hamiltonians that Poisson-commute with a Tonelli Hamiltonian.

On commuting Tonelli Hamiltonians: Autonomous case

We show that the Aubry sets, the Mañé sets and the barrier functions are the same for two commuting autonomous Tonelli Hamiltonians. We also show the quasi-linearity of α-functions from the viewpoint of dynamics and the existence of common C1,1 critical subsolution for their associated Hamilton–Jacobi equations.

On the stability of Mañé critical hypersurfaces

We construct examples of Tonelli Hamiltonians on \({\mathbb{T}^n}\) (for any n ≥ 2) such that the hypersurfaces corresponding to the Mane critical value are stable (i.e. geodesible). We also provide a criterion for instability in terms of closed orbits in free homotopy classes and we show that any stable energy level of a Tonelli Hamiltonian must contain a closed orbit.

Weak KAM for commuting Hamiltonians

For two commuting Tonelli Hamiltonians, we recover the commutation of the Lax–Oleinik semi-groups, a result of Barles and Tourin (2001 Indiana Univ. Math. J. 50 1523–44), using a direct geometrical method (Stoke's theorem). We also obtain a ‘generalization’ of a theorem of Maderna (2002 Bull. Soc. Math. France 130 493–506). More precisely, we prove that if the phase space is the cotangent of a compact manifold then the weak KAM solutions (or viscosity solutions of the critical stationary Hamilton–Jacobi equation) for G and for H are the same. As a corollary we obtain the equality of the Aubry sets and of the Peierls barrier. This is also related to works of Sorrentino (2009 On the Integrability of Tonelli Hamiltonians Preprint) and Bernard (2007 Duke Math. J. 136 401–20).