Related papers: On commuting Tonelli Hamiltonians: Time-periodic c…

An old conjecture claims that commuting Hamiltonians of the double-elliptic integrable system are constructed from the theta-functions associated with Riemann surfaces from the Seiberg-Witten family, with moduli treated as dynamical…

We prove the existence of at least $cl(M)$ periodic orbits for certain time dependant Hamiltonian systems on the cotangent bundle of an arbitrary compact manifold $M$. These Hamiltonians are not necessarily convex but they satisfy a certain…

Let $M$ be a closed and connected manifold, $H:T^*M\times \mathbb{R} / \mathbb{Z} \to \mathbb{R}$ a Tonelli $1$-periodic Hamiltonian and $\mathcal{L} \subset T^*M$ a Lagrangian submanifold Hamiltonianly isotopic to the zero section. We…

Let $M$ be a closed manifold and consider the Hamiltonian flow associated to an autonomous Tonelli Hamiltonian $H:T^*M\rightarrow \mathbb R$ and a twisted symplectic form. In this paper we study the existence of contractible periodic orbits…

An example due to Pisier shows that two commuting, completely polynomially bounded Hilbert space operators may not be simultaneously similar to contractions. Thus, while each operator is individually similar to a contraction, the pair is…

The U(1) Calogero-Sutherland Model with anti-periodic boundary condition is studied. This model is obtained by applying a vertical magnetic field perpendicular to the plane of one dimensional ring of particles. The trigonometric form of the…

We prove a perturbation theorem for the $k$-jets, $k\geq 2$, of the Poincar\'e map of a closed orbit of the Hamiltonian flow of a Tonelli Hamiltonian $H: T^*M\to \R$, on a closed manifold $M$. As a consequence we obtain Ma\~n\'e generic…

We prove that a finite family of commuting holomorphic self-maps of the unit ball $\mathbb{B}^q\subset \mathbb{C}^q$ admits a simultaneous holomorphic conjugacy to a family of commuting automorphisms of a possibly lower dimensional ball,…

It has been known since Julia that polynomials commuting under composition have the same Julia set. More recently in the works of Baker and Eremenko, Fern\'andez, and Beardon, results were given on the converse question: When do two…

We investigate the existence of Arnold diffusion-type orbits for systems obtained by iterating in any order the time-one maps of a family of Tonelli Hamiltonians. Such systems are known as 'polysystems' or 'iterated function systems'. When…

We investigate to what extent a minimal topological dynamical system is uniquely determined by a set of return times to some open set. We show that in many situations this is indeed the case as long as the closure of this open set has no…

We prove the equivalence between anyon quantum mechanics on a torus and Chern-Simons gauge theory. It is also shown that the Hamiltonian and total momenta commute among themselves only in the physical Hilbert space.

We prove that $C^2$ generic hyperbolic Ma\~n\'e sets contain a periodic periodic orbit. In dimension 2, adding a result by Contreras, Figalli, Rifford, which states that $C^2$ generic Ma\~n\'e sets are hyperbolic; we obtain Ma\~n\'e's…

The goal of this paper is to find the quantization conditions of Bohr-Sommerfeld of k quantum Hamiltonians acting on the euclidian space of dimension n, depending on a small parameter h, and which commute to each other. That is we…

The equivalence of two complete sets of Poisson commuting Hamiltonians of the (super)integrable rational BC(n) Ruijsenaars-Schneider-van Diejen system is established. Specifically, the commuting Hamiltonians constructed by van Diejen are…

It is shown that when the gauge algebra is with root system the canonical Hamiltonian commutes with the constraints. Two other simple propositions concerning gauge fixing are proved too.

It is shown that sl(2,$\mathbb{C}$) Euler-Arnold tops are equivalent to the two-body systems of Calogero-Moser type. We prove that generic Hamiltonians of sl(2,$\mathbb{C}$) tops are equivalent to one of three canonical Hamiltonians. For…

We show that a set with an action of a locally finite-dimensional free partially commutative monoid and the corresponding semicubical set have isomorpic homology groups. We build a complex of finite length for the computing homology groups…

The classical theorem of Moser, on the existence of a normal form in the neighbourhood of a hyperbolic equilibrium, is extended to a class of real-analytic Hamiltonians with aperiodically time-dependent perturbations. A stronger result is…

We construct two series of commuting Hamiltonians parametrized by a constant matrix. The first series was my guess and the second one was know, and in our approach follows from the first series. For the proof we use familier facts known…