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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.
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
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
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 this article we show that Bers simultaneous uniformization as well as the Koebes retrosection theorem are not longer true for discrete groups of projective transformations acting on the complex projective space.
The counting and (upper) mass dimensions are notions of dimension for subsets of $mathbb{Z}^d$. We develop their basic properties and give a characterization of the counting dimension via coverings. In addition, we prove Marstrand-type results for bo