For any graded poset $P$, we define a new graded poset, $mathcal E(P)$, whose elements are the edges in the Hasse diagram of P. For any group, $G$, acting on the boolean algebra, $B_n$, we conjecture that $mathcal E(B_n/G)$ is Peck. We prove that the
conjecture holds for common cover transitive actions. We give some infinite families of common cover transitive actions and show that the common cover transitive actions are closed under direct and semidirect products.
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 clust
er 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.
In this paper, we discuss rotation number on the invariant curve of a one parameter family of outer billiard tables. Given a convex polygon $eta$, we can construct an outer billiard table $T$ by cutting out a fixed area from the interior of $eta$. $T
$ is piece-wise hyperbolic and the polygon $eta$ is an invariant curve of $T$ under the billiard map $phi$. We will show that, if $beta $ is a periodic point under the outer billiard map with rational rotation number $tau = p / q$, then the $n$th iteration of the billiard map is not the local identity at $beta$. This proves that the rotation number $tau$ as a function of the area parameter is a devils staircase function.
