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Register a new userThe Pentagram map in higher dimensions and KdV flows
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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.
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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. The corresponding continuous limit of the pentagram map in dimension $d$ is shown to be the $(2,d+1)$-equation of the KdV hierarchy, generalizing the Boussinesq equation in 2D. We also study in detail the 3D case, where we prove integrability for both closed and twisted polygons and describe the spectral curve, first integrals, the corresponding tori and the motion along them, as well as an invariant symplectic structure.
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 energy level, there is a waist on every energy level just above the Ma~ne critical value. With a suitable perturbation with a potential, we show that there are infinitely many periodic orbits on every energy level just above the Ma~ne critical value, and on almost every energy level just below. Finally, we prove the Tonelli analogue of a closed geodesics result due to Ballmann-Thorbergsson-Ziller.
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 that those integrable magnetic systems admitting a global surface of section satisfy a sharp systolic inequality.
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 in the case of isometries of CAT(0)-spaces, can be given by means of tree three types, namely: elliptic, parabolic and loxodromic. We carefully describe the dynamic in each case, more precisely we determine the corresponding Kulkarnis limit set, the equicontinuity region, the discontinuity region and in some cases we provide families of maximal regions where the respective cyclic group acts properly discontinuously. Also we provide, in each case, some equivalents ways to classify the projective transformations.
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 for orbits connecting two given closed submanifolds of $M$ satisfying the conormal boundary conditions: We introduce the Ma~ne critical value that is relevant for the problem and prove existence results for supercritical and subcritical energies; we also complement these with counterexamples, thus showing the sharpness of our results. We then move to the problem of finding periodic orbits: We provide an existence result of periodic orbits for non-aspherical manifolds generalizing the Lusternik-Fet Theorem, and a multiplicity result in case the configuration space is the 2-torus.
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