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We extend the definition of the pentagram map from 2D to higher dimensions and describe its integrability properties for both closed and twisted polygons by presenting its Lax form. The corresponding continuous limit of the pentagram map in dimension $d$ is shown to be the $(2,d+1)$-flow of the KdV hierarchy, generalizing the Boussinesq equation in 2D.
We define higher pentagram maps on polygons in $P^d$ for any dimension $d$, which extend R.Schwartzs definition of the 2D pentagram map. We prove their integrability by presenting Lax representations with a spectral parameter for scale invariant maps
We study the periodic orbits problem on energy levels of Tonelli Lagrangian systems over configuration spaces of arbitrary dimension. We show that, when the fundamental group is finite and the Lagrangian has no stationary orbit at the Ma~ne critical
In this paper we study some aspects of integrable magnetic systems on the two-torus. On the one hand, we construct the first non-trivial examples with the property that all magnetic geodesics with unit speed are closed. On the other hand, we show tha
In this article we provide a classification of the projective transformations in $PSL(n+1,Bbb{C})$ considered as automorphisms of the complex projective space $Bbb{P}^n$. Our classification is an interplay between algebra and dynamics, which just as
Let $(M,g)$ be a closed Riemannian manifold and $L:TMrightarrow mathbb R$ be a Tonelli Lagrangian. In this thesis we study the existence of orbits of the Euler-Lagrange flow associated with $L$ satisfying suitable boundary conditions. We first look f