arXiv subjects

Skander Charfi

Publications and source records attributed to Skander Charfi.

4 recordsLinked to original sources

arXiv2026

A Multidimensional Birkhoff Theorem for some $C^0$ Lagrangians

We prove a multidimensional Birkhoff theorem for a new class of "$C^0$ Lagrangian subsets" in cotangent bundles, obtained as Hausdorff limits of compact exact Lagrangian submanifolds with control on their Liouville primitives. If the successive images of such a subset $L$ under the flow of a Tonelli Hamiltonian admit convergent subsequences in both positive and negative time, then $L$ and all its images are Lipschitz graphs over the base. This extends Birkhoff's celebrated theorem for twist maps of the annulus and its known higher-dimensional generalizations, and provides a first version of such results for merely continuous objects. The proof combines Floer-theoretic graph selectors, variational solutions of the Hamilton-Jacobi equation, and weak KAM theory. We also investigate the rigidity and uniqueness of limiting primitives for this new class of singular Lagrangian subsets, which may be of independent interest in $C^0$ symplectic topology.

math.SG↗

arXiv2025

A Multidimensional Birkhoff Theorem for Recurrent Lagrangian Submanifolds by a Tonelli Hamiltonian

Consider a closed manifold $M$ and a time-periodic Tonelli Hamiltonian $H : \mathbb{R}/\mathbb{Z} \times T^*M \to \mathbb{R}$ with flow $ϕ_H$. Let $\mathcal{L} \subset T^*M$ be a Lagrangian submanifold Hamiltonianly isotopic to the zero section. We prove that if $ϕ_H^n(\mathcal{L})$ admits convergent subsequences in both positive and negative times, in the Hausdorff topology and with control on the Liouville primitives, to two Lagrangian submanifolds, then $\mathcal{L}$ is a graph over the zero section $0_{T^*M}$ of $T^*M$. Furthermore, we show that $\mathcal{L}$ is recurrent in both positive and negative times for the same type of convergence.

math.DS↗

arXiv2025

A Smooth, Recurrent, Non-Periodic Viscosity Solution of the Hamilton-Jacobi Equation

Viscosity solutions of the Hamilton-Jacobi equation were introduced by Lions and Crandall. For Tonelli Hamiltonians, these solutions are generated by the Lax-Oleinik operator. It is known that this operator converges in the autonomous framework, but this convergence fails in the general cases. In this paper, we introduce a method to construct smooth, recurrent, non-periodic viscosity solutions on fixed compact manifolds $M$ of dimension 2 or higher. Additionally, we provide a detailed description of the non-wandering set of the Lax-Oleinik operator and identify its action on various omega-limit sets.

math.DS↗

arXiv2025

Representation of Global Viscosity Solutions for Tonelli Hamiltonians

We consider the Lax-Oleinik operator $\mathcal{T}$ associated with the non-stationary Hamilton-Jacobi equation $\partial_tu + H(t,x,\partial_xu) = α_0$ for a Tonelli Hamiltonian $H$ and its \Mane critical value $α_0$. It is known from the work of A. Fathi and J.N. Mather \cite{MR1792479} that the convergence of this semigroup fails in the non-autonomous framework. In this context, we study the action of $\mathcal{T}$ on its non-wandering set $Ω(\mathcal{T})$. First, we show that $\mathcal{T}$ acts as an isometry on this set, and then we characterize $Ω(\mathcal{T})$ as the set of global viscosity solutions of the Hamilton-Jacobi equation, i.e. solutions that are defined for all real times. Next, we introduce a generalized Peierls barrier $\underline{k}$ and a set of generalized static classes $\underline{\mathbb{M}}$ within the Mather set. Using these, we represent elements $u$ of $Ω(\mathcal{T})$ as \begin{equation*} u(x) = \inf_{y \in \underline{\mathbb{M}}} \{ u(y) + \underline{k}(y,x) \} \end{equation*} We apply this representation formula to prove Fathi's convergence theorem for autonomous systems and provide a representation formula for $n$-periodic viscosity solutions. Additionally, we establish that the dynamics of non-wandering viscosity solutions are governed by the Lagrangian flow on the Mather set. Specifically, we show that if the Mather set consists solely of $N$-periodic orbits for some integer $N$, then all non-wandering viscosity solutions are $N$-periodic. Furthermore, we show that if the restriction of the Lagrangian flow to the Mather set is uniformly recurrent for a time sequence $p_n$, then all non-wandering viscosity solutions are uniformly recurrent for the same time sequence $p_n$.

math.DS↗