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Register a new userPoncelet pairs and the Twist Map associated to the Poncelet Billiard
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We show that for a fixed curve $K$ and for a family of variables curves $L$, the number of $n$-Poncelet pairs is $frac{e (n)}{2}$, where $e(n)$ is the number of natural numbers $m$ smaller than $n$ and which satisfies mcd $ (m,n)=1$. The curvee $K$ do not have to be part of the family. In order to show this result we consider an associated billiard transformation and a twist map which preserves area. We use Aubry-Mather theory and the rotation number of invariant curves to obtain our main result. In the last section we estimate the derivative of the rotation number of a general twist map using some properties of the continued fraction expansion .
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In this note I provide two extensions of a particular case of the classical Poncelet theorem.
We analyze loci of triangles centers over variants of two-well known triangle porisms: the bicentric family and the confocal family. Specifically, we evoke a more general version of Poncelets closure theorem whereby individual sides can be made tangent to separate caustics. We show that despite a more complicated dynamic geometry, the locus of certain triangle centers and associated points remain conics and/or circles.
In this paper we present an upper bound for the decay of correlation for the stationary stochastic process associated with the Entropy Penalized Method. Let $L(x, v):Tt^ntimesRr^nto Rr$ be a Lagrangian of the form L(x,v) = {1/2}|v|^2 - U(x) + < P, v>. For each value of $epsilon $ and $h$, consider the operator Gg[phi](x):= -epsilon h {ln}[int_{re^N} e ^{-frac{hL(x,v)+phi(x+hv)}{epsilon h}}dv], as well as the reversed operator bar Gg[phi](x):= -epsilon h {ln}[int_{re^N} e^{-frac{hL(x+hv,-v)+phi(x+hv)}{epsilon h}}dv], both acting on continuous functions $phi:Tt^nto Rr$. Denote by $phi_{epsilon,h} $ the solution of $Gg[phi_{epsilon,h}]=phi_{epsilon,h}+lambda_{epsilon,h}$, and by $bar phi_{epsilon,h} $ the solution of $bar Gg[phi_{epsilon,h}]=bar phi_{epsilon,h}+lambda_{epsilon,h}$. In order to analyze the decay of correlation for this process we show that the operator $ {cal L} (phi) (x) = int e^{- frac{h L (x,v)}{epsilon}} phi(x+h v) d v,$ has a maximal eigenvalue isolated from the rest of the spectrum.
The action of a Backlund-Darboux transformation on a spectral problem associated with a known integrable system can define a new discrete spectral problem. In this paper, we interpret a slightly generalized version of the binary Backlund-Darboux (or Zakharov-Shabat dressing) transformation for the nonlinear Schrodinger (NLS) hierarchy as a discrete spectral problem, wherein the two intermediate potentials appearing in the Darboux matrix are considered as a pair of new dependent variables. Then, we associate the discrete spectral problem with a suitable isospectral time-evolution equation, which forms the Lax-pair representation for a space-discrete NLS system. This formulation is valid for the most general case where the two dependent variables take values in (rectangular) matrices. In contrast to the matrix generalization of the Ablowitz-Ladik lattice, our discretization has a rational nonlinearity and admits a Hermitian conjugation reduction between the two dependent variables. Thus, a new proper space-discretization of the vector/matrix NLS equation is obtained; by changing the time part of the Lax pair, we also obtain an integrable space-discretization of the vector/matrix modified KdV (mKdV) equation. Because Backlund-Darboux transformations are permutable, we can increase the number of discrete independent variables in a multi-dimensionally consistent way. By solving the consistency condition on the two-dimensional lattice, we obtain a new Yang-Baxter map of the NLS type, which can be considered as a fully discrete analog of the principal chiral model for projection matrices.
We consider the Laplacian with a delta potential (a point scatterer) on an irrational torus, where the square of the side ratio is diophantine. The eigenfunctions fall into two classes ---old eigenfunctions (75%) of the Laplacian which vanish at the support of the delta potential, and therefore are not affected, and new eigenfunctions (25%) which are affected, and as a result feature a logarithmic singularity at the location of the delta potential. Within a full density subsequence of the new eigenfunctions we determine all semiclassical measures in the weak coupling regime and show that they are localized along 4 wave vectors in momentum space --- we therefore prove the existence of so-called superscars as predicted by Bogomolny and Schmit. This result contrasts the phase space equidistribution which is observed for a full density subset of the new eigenfunctions of a point scatterer on a rational torus. Further, in the strong coupling limit we show that a weaker form of localization holds for a positive proportion of the new eigenvalues; in particular quantum ergodicity does not hold. We also explain how our results can be modified for rectangles with Dirichlet boundary conditions with a point scatterer in the interior. In this case our results extend previous work of Keating, Marklof and Winn who proved the existence of localized semiclassical measures under a non-clustering condition on the spectrum of the Laplacian.
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