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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.
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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 $d$ is shown to be the $(2,d+1)$-flow of the KdV hierarchy, generalizing the Boussinesq equation in 2D.
For a tuple $A=(A_1, A_2, ..., A_n)$ of elements in a unital algebra ${mathcal B}$ over $mathbb{C}$, its {em projective spectrum} $P(A)$ or $p(A)$ is the collection of $zin mathbb{C}^n$, or respectively $zin mathbb{P}^{n-1}$ such that the multi-parameter pencil $A(z)=z_1A_1+z_2A_2+cdots +z_nA_n$ is not invertible in ${mathcal B}$. ${mathcal B}$-valued $1$-form $A^{-1}(z)dA(z)$ contains much topological information about $P^c(A):=mathbb{C}^nsetminus P(A)$. In commutative cases, invariant multi-linear functionals are effective tools to extract that information. This paper shows that in non-commutative cases, the cyclic cohomology of ${mathcal B}$ does a similar job. In fact, a Chen-Weil type map $kappa$ from the cyclic cohomology of ${mathcal B}$ to the de Rham cohomology $H^*_d(P^c(A), mathbb{C})$ is established. As an example, we prove a closed high-order form of the classical Jacobis formula.
In a recent paper, Teo and Kane proposed a 3D model in which the defects support Majorana fermion zero modes. They argued that exchanging and twisting these defects would implement a set R of unitary transformations on the zero mode Hilbert space which is a ghostly recollection of the action of the braid group on Ising anyons in 2D. In this paper, we find the group T_{2n} which governs the statistics of these defects by analyzing the topology of the space K_{2n} of configurations of 2n defects in a slowly spatially-varying gapped free fermion Hamiltonian: T_{2n}equiv {pi_1}(K_{2n})$. We find that the group T_{2n}= Z times T^r_{2n}, where the ribbon permutation group T^r_{2n} is a mild enhancement of the permutation group S_{2n}: T^r_{2n} equiv Z_2 times E((Z_2)^{2n}rtimes S_{2n}). Here, E((Z_2)^{2n}rtimes S_{2n}) is the even part of (Z_2)^{2n} rtimes S_{2n}, namely those elements for which the total parity of the element in (Z_2)^{2n} added to the parity of the permutation is even. Surprisingly, R is only a projective representation of T_{2n}, a possibility proposed by Wilczek. Thus, Teo and Kanes defects realize `Projective Ribbon Permutation Statistics, which we show to be consistent with locality. We extend this phenomenon to other dimensions, co-dimensions, and symmetry classes. Since it is an essential input for our calculation, we review the topological classification of gapped free fermion systems and its relation to Bott periodicity.
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.
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