We consider the cubic defocusing nonlinear Schrödinger equation in the two dimensional torus. Fix s > 1. Colliander, Keel, Staffilani, Tao and Takaoka proved in [CKS10] the existence of solutions… (More)

We consider the image of a fractal set X in a Banach space under typical linear and nonlinear projections π intoR . We prove that whenN exceeds twice the box-counting dimension of X, then almost… (More)

We introduce a new potential-theoretic definition of the dimension spectrum Dq of a probability measure for q > 1 and explain its relation to prior definitions. We apply this definition to prove that… (More)

The Existential Hilbert Problem is a weak version of the part b of the Hilbert 16-th problem which also asks not only about the number, but also about position of limit cycles of (1). The problem… (More)

Let M be a smooth compact manifold of dimension at least 2 and Diff (M) be the space of C smooth diffeomorphisms of M . Associate to each diffeomorphism f ∈ Diff (M) the sequence Pn(f ) of the number… (More)

In the present paper we prove a form of Arnold diffusion. The main result says that for a ”generic” perturbation of a nearly integrable system of arbitrary degrees of freedom n > 2 H0(p) + εH1(θ, p,… (More)

A stratification of a set, e.g. an analytic variety, is, roughly, a partition of it into manifolds so that these manifolds fit together “regularly”. Stratification theory was originated by Thom and… (More)

In this paper we present simplified proofs of two important theorems of J.Mather. The first (connecting) theorem [Ma2] is about wandering trajectories of exact area-preserving twist maps naturally… (More)