Alfonso Sorrentino

On the integrability of Tonelli Hamiltonians

In this article we discuss a weaker version of Liouvilles 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-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 one.

Differentiability of Mather's average action and integrability on closed surfaces

In this article we study the differentiability of Mathers

Weak Liouville-Arnol′d Theorems and Their Implications

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Action minimizing properties and distances on the group of Hamiltonian diffeomorphisms

-function on closed surfaces and its relation to the integrability of the system.

On the total disconnectedness of the quotient Aubry set

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-Arnol′d 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.

Aubry–Mather Theory for Conformally Symplectic Systems

In this article we prove that for a smooth fiberwise convex Hamiltonian, the asymptotic Hofer distance from the identity gives a strict upper bound to the value at 0 of Mather’s function, thus providing a negative answer to a question asked by Siburg [15]. However, we show that equality holds if one considers the asymptotic distance defined by Viterbo [20]. 37J05, 37J50; 53D35

On the marked length spectrum of generic strictly convex billiard tables

In this paper we show that the quotient Aubry set associated to certain Lagrangians is totally disconnected (i.e., every connected component consists of a single point). Moreover, we discuss the relation between this problem and a Morse-Sard type property for (difference of) critical subsolutions of Hamilton-Jacobi equations.

On the integrability of Birkhoff billiards

In this article we develop an analogue of Aubry–Mather theory for a class of dissipative systems, namely conformally symplectic systems, and prove the existence of interesting invariant sets, which, in analogy to the conservative case, will be called the Aubry and the Mather sets. Besides describing their structure and their dynamical significance, we shall analyze their attracting/repelling properties, as well as their noteworthy role in driving the asymptotic dynamics of the system.

Global results for eikonal Hamilton–Jacobi equations on networks

In this paper we show that for a generic strictly convex domain, one can recover the eigendata corresponding to Aubry–Mather periodic orbits of the induced billiard map from the (maximal) marked length spectrum of the domain.

Lecture notes on Mather's theory for Lagrangian systems

In this survey, we provide a concise introduction to convex billiards and describe some recent results, obtained by the authors and collaborators, on the classification of integrable billiards, namely the so-called Birkhoff conjecture. This article is part of the theme issue ‘Finite dimensional integrable systems: new trends and methods’.