Absolutely continuous cocycles over irrational rotations

We prove global well-posedness for the Gross-Pitaevskii equation on the plane for classes of initial data having nonzero topological degree at infinity and therefore infinite Ginzburg-Landau energy. These classes allow us to consider arbitrary configurations of vortices as initial data. Our work follows recent results of Patrick Gérard [9] and Clément Gallo [4], where the finite energy regime is treated.

In this paper we give a simple, explicit construction of polytopal bounded remainder sets of all possible volumes, for any irrational rotation on the $d$ dimensional adelic torus $\mathbb{A}^d/\mathbb{Q}^d$. Our construction involves ideas from dynamical systems and harmonic analysis on the adeles, as well as a geometric argument that reduces the existence argument to the case of an irrational rotation on the torus $\mathbb{R}^d/\mathbb{Q}^d$.

Persistence of lower dimensional hyperbolic tori for reversible system

A weighted complementarity problem is to find a pair of vectors belonging to the intersection of a manifold and a cone such that the product of the vectors in a certain algebra equals a given weight vector. If the weight vector is zero, we get a complementarity problem. Examples of such problems include the Fisher market equilibrium problem and the linear programming and weighted centering problem. In this paper we consider the weighted horizontal linear complementarity problem in the setting of Euclidean Jordan algebras and establish some existence and uniqueness results. For a pair of linear transformations on a Euclidean Jordan algebra, we introduce the concepts of $$\mathbf{R}_0$$ , $$\mathbf{R}$$ , and $$\mathbf{P}$$ properties and discuss the solvability of wHLCPs under nonzero (topological) degree conditions. A uniqueness result is stated in the setting of $${\mathbb {R}}^{n}$$ . We show how our results naturally lead to interior point systems.

The maximally chaotic K-systems are dynamical systems which have nonzero Kolmogorov entropy. On the other hand, the hyperbolic dynamical systems that fulfil the Anosov C-condition have exponential instability of phase trajectories, mixing of all orders, countable Lebesgue spectrum and positive Kolmogorov entropy. The C-condition defines a rich class of maximally chaotic systems which span an open set in the space of all dynamical systems. The interest in Anosov-Kolmogorov C-K systems is associated with the attempts to understand the relaxation phenomena, the foundation of the statistical mechanics, the appearance of turbulence in fluid dynamics, the non-linear dynamics of the Yang-Mills field as well as the dynamical properties of gravitating N-body systems and the Black hole thermodynamics. In this respect of special interest are C-K systems that are defined on Reimannian manifolds of negative sectional curvature and on a high-dimensional tori. Here we shall review the classical- and quantum-mechanical properties of maximally chaotic dynamical systems, the application of the C-K theory to the investigation of the Yang-Mills dynamics and gravitational systems as well as their application in the Monte Carlo method.

Exponential convergence of the weighted Birkhoff average

In this paper, we consider a classical Hamiltonian normal form with degeneracy in the normal direction. In previous results, one needs to assume that the perturbation satisfies certain non-degenerate conditions in order to remove the degeneracy in the normal form. Instead of that, we introduce a topological degree condition and a weak convexity condition, which are easy to verify, and we prove the persistence of lower dimensional tori without any restriction on perturbation but only smallness and analyticity.