Non-differentiable irrational curves for twist map

We construct a $C^1$ symplectic twist map g of the annulus that has an essential invariant curve $\Gamma $ such that $\Gamma $ is not differentiable and g restricted to $\Gamma $ is minimal.

Polyadic Hopf Algebras and Quantum Groups

This article continues the study of concrete algebra-like structures in our polyadic approach, where the arities of all operations are initially taken as arbitrary, but the relations between them, the arity shapes, are to be found from some natural conditions (“arity freedom principle”). In this way, generalized associative algebras, coassociative coalgebras, bialgebras and Hopf algebras are defined and investigated. They have many unusual features in comparison with the binary case. For instance, both the algebra and its underlying field can be zeroless and nonunital, the existence of the unit and counit is not obligatory, and the dimension of the algebra is not arbitrary, but “quantized”. The polyadic convolution product and bialgebra can be defined, and when the algebra and coalgebra have unequal arities, the polyadic version of the antipode, the querantipode, has different properties. As a possible application to quantum group theory, we introduce the polyadic version of braidings, almost co-commutativity, quasitriangularity and the equations for the R-matrix (which can be treated as a polyadic analog of the Yang-Baxter equation). We propose another concept of deformation which is governed not by the twist map, but by the medial map, where only the latter is unique in the polyadic case. We present the corresponding braidings, almost co-mediality and M-matrix, for which the compatibility equations are found.

Parity duality for the amplituhedron

The (tree) amplituhedron $\mathcal {A}_{n,k,m}(Z)$ is a certain subset of the Grassmannian introduced by Arkani-Hamed and Trnka in 2013 in order to study scattering amplitudes in $N=4$ supersymmetric Yang–Mills theory. Confirming a conjecture of the first author, we show that when $m$ is even, a collection of affine permutations yields a triangulation of $\mathcal {A}_{n,k,m}(Z)$ for any $Z\in \operatorname {Gr}_{>0}(k+m,n)$ if and only if the collection of their inverses yields a triangulation of $\mathcal {A}_{n,n-m-k,m}(Z)$ for any $Z\in \operatorname {Gr}_{>0}(n-k,n)$. We prove this duality using the twist map of Marsh and Scott. We also show that this map preserves the canonical differential forms associated with the corresponding positroid cells, and hence obtain a parity duality for amplituhedron differential forms.

The twist for positroids

International audience There are two reasonable ways to put a cluster structure on a positroid variety. In one, the initial seed is a set of Plu ̈cker coordinates. In the other, the initial seed consists of certain monomials in the edge weights of a plabic graph. We will describe an automorphism of the positroid variety, the twist, which takes one to the other. For the big positroid cell, this was already done by Marsh and Scott; we generalize their results to all positroid varieties. This also provides an inversion of the boundary measurement map which is more general than Talaska's, in that it works for all reduced plabic graphs rather than just Le-diagrams. This is the analogue for positroid varieties of the twist map of Berenstein, Fomin and Zelevinsky for double Bruhat cells. Our construction involved the combinatorics of dimer configurations on bipartite planar graphs.

The integrability of symplectic twist maps without conjugate points

It is proved that a symplectic twist map of the cotangent bundle $T^{\ast }\mathbb{T}^{d}$ of the $d$-dimensional torus that is without conjugate points is $C^{0}$-integrable, that is $T^{\ast }\mathbb{T}^{d}$ is foliated by a family of invariant $C^{0}$ Lagrangian graphs.

Prime factors of quantum Schubert cell algebras and clusters for quantum Richardson varieties

The understanding of the topology of the spectra of quantum Schubert cell algebras hinges on the description of their prime factors by ideals invariant under the maximal torus of the ambient Kac–Moody group. We give an explicit description of these prime quotients by expressing their Cauchon generators in terms of sequences of normal elements in chains of subalgebras. Based on this, we construct large families of quantum clusters for all of these algebras and the quantum Richardson varieties associated to arbitrary symmetrizable Kac–Moody algebras and all pairs of Weyl group elements. Along the way we develop a quantum version of the Fomin–Zelevinsky twist map for all quantum Richardson varieties. Furthermore, we establish an explicit relationship between the Goodearl–Letzter and Cauchon approaches to the descriptions of the spectra of symmetric CGL extensions.

Area-preserving diffeomorphisms

This chapter is devoted to two-dimensional symplectomorphisms, which are just area- and orientation-preserving diffeomorphisms. The chapter includes an exposition of Birkhoff’s proof of Poincaré’s last geometric theorem, which asserts that an area-preserving twist map of the annulus must have at least 4 two distinct fixed points. It also discusses some applications of these ideas to billiard problems.

On reversible maps and symmetric periodic points

In reversible dynamical systems, it is of great importance to understand symmetric features. The aim of this paper is to explore symmetric periodic points of reversible maps on planar domains invariant under a reflection. We extend Franks’ theorem on a dichotomy of the number of periodic points of area-preserving maps on the annulus to symmetric periodic points of area-preserving reversible maps. Interestingly, even a non-symmetric periodic point guarantees infinitely many symmetric periodic points. We prove an analogous statement for symmetric odd-periodic points of area-preserving reversible maps isotopic to the identity, which can be applied to dynamical systems with double symmetries. Our approach is simple, elementary, and far from Franks’ proof. We also show that a reversible map has a symmetric fixed point if and only if it is a twist map which generalizes a boundary twist condition on the closed annulus in the sense of Poincaré–Birkhoff. Applications to symmetric periodic orbits in reversible dynamical systems with two degrees of freedom are briefly discussed.