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The pentagram map has been studied in a series of papers by Schwartz and others. Schwartz showed that an axis-aligned polygon collapses to a point under a predictable number of iterations of the pentagram map. Glick gave a different proof using cluster algebras, and conjectured that the point of collapse is always the center of mass of the axis-aligned polygon. In this paper, we answer Glicks conjecture positively, and generalize the statement to higher and lower dimensional pentagram maps. For the latter map, we define a new system -- the mirror pentagram map -- and prove a closely related result. In addition, the mirror pentagram map provides a geometric description for the lower dimensional pentagram map, defined algebraically by Gekhtman, Shapiro, Tabachnikov and Vainshtein.
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
Let $V$ be a Banach space where for fixed $n$, $1<n<dim(V)$, all of its $n$-dimensional subspaces are isometric. In 1932, Banach asked if under this hypothesis $V$ is necessarily a Hilbert space. Gromov, in 1967, answered it positively for even $n$ a
Let E be a compact set in the plane, g be a K-quasiconformal map, and let 0<t<2. Then H^t (E) = 0 implies H^{t} (g E) = 0, for t=[2Kt]/[2+(K-1)t]. This is a refinement of a set of inequalities on the distortion of Hausdorff dimensions by quasicon
We show that for a monic polynomial $f(x)$ over a number field $K$ containing a global permutation polynomial of degree $>1$ as its composition factor, the Newton Polygon of $fmodmathfrak p$ does not converge for $mathfrak p$ passing through all fini
We show that a jammed packing of disks with generic radii, in a generic container, is such that the minimal number of contacts occurs and there is only one dimension of equilibrium stresses. We also point out some connections to packings with differe