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.
We investigate the high temperature phase of layered manganites, and demonstrate that the charge-orbital phase transition without magnetic order in La$_{0.5}$Sr$_{1.5}$MnO$_4$ can be understood in terms of the density wave instability. The orbital or
dering is found to be induced by the nesting between segments of Fermi surface with different orbital characters. The simultaneous charge and orbital orderings are elaborated with a mean field theory. The ordered orbitals are shown to be $d_{x^2-y^2} pm d_{3z^2-r^2}$.
We report theoretical and experimental studies of the effect of Zn-impurity in Fe-based superconductors. Zn-impurity is expected to severely suppress sign reversed s$_pm$ wave pairing. The experimentally observed suppression of T$_c$ under Zn-doping
strongly depends on the materials and the charge carrier contents, which suggests competition of $s_{++}$ and $s_{pm}$ pairings in Fe-base superconductors. We study a model incorporating both $s_{++}$ and $s_{pm}$ pairing couplings by using Bogoliubov de-Gennes equation, and show that the Zn-impurity strongly suppresses $s_{pm}$ pairing and may induce a transition from $s_{pm}$ to $s_{++}$-wave. Our theory is consistent with various experiments on the impurity effect. We present new experimental data on the Zn-doping SmFe$_{1-x}$Zn$_x$AsO$_{0.9}$F$_{0.1}$ of T$_c=$ 50K, in further support of our proposal.
Based on an effective two-band model and using the fluctuation-exchange (FLEX) approach, we explore spin fluctuations and unconventional superconducting pairing in Fe-based layer superconductors. It is elaborated that one type of interband antiferrom
agnetic (AF) spin fluctuation stems from the interband Coulomb repulsion, while the other type of intraband AF spin fluctuation originates from the intraband Coulomb repulsion. Due to the Fermi-surface topology, a spin-singlet extended s-wave superconducting state is more favorable than the nodal $d_{XY}$-wave state if the interband AF spin fluctuation is more significant than the intraband one, otherwise vice versa. It is also revealed that the effective interband coupling plays an important role in the intraband pairings, which is a distinct feature of the present two-band system.
Based on a single band Hubbard model and the fluctuation exchange approximation, the effective mass and the energy band renormalization in Na$_{0.33}$CoO$_2$ is elaborated. The renormalization is observed to exhibit certain kind of anisotropy, which
agrees qualitatively with the angle-resolved photoemission spectroscopy (ARPES) measurements. Moreover, the spectral function and density of states (DOS) in the normal state are calculated, with a weak pseudogap behavior being seen, which is explained as a result of the strong Coulomb correlations. Our results suggest that the large Fermi surface (FS) associated with the $a_{1g}$ band plays likely a central role in the charge dynamics.
