Almost periodic sets and measures on the torus

We establish some “number theoretical” results about a continuous functionh from the circleT into itself, which generalize Kronecker’s theorem in several ways. These results are used to characterize the almost periodic sets of the flow on the torusT ² generated by (θ, φ) → (θ+α, φ+h(θ)), where α is irrational. The almost periodic measures are characterized in the caseh(θ)=θ.     

Almost periodic hyperfunctions

We characterize the almost periodic hyperfunctions by showing that the following statements are equivalent for any bounded hyperfunction . (i) is almost periodic. (ii) for every . (iii) There are two functions and an infinite order differential operator such that (iv) The Gauss transform of is almost periodic for every 0$">. Here is the space of almost periodic continuous functions, is the Sato space of test functions for the Fourier hyperfunctions, and is the heat kernel. This generalizes the result of Schwartz on almost periodic distributions and that of Cioranescu on almost periodic (non-quasianalytic) ultradistributions to the case of hyperfunctions.

     

Almost periodic Harmonizable processes

The class of harmonizable processes is a natural extension of the class of stationary processes. This paper provides sufficient conditions for the sample paths of harmonizable processes to be almost periodic uniformly, Stepanov and Besicovitch.     

Almost periodic fuzzy-number-valued functions

In this paper we develop a theory of almost periodic fuzzy functions, i.e. of the almost periodic functions of real variable and with values fuzzy real numbers, Although the class of fuzzy real numbers does not form a linear normed space, the majority of main properties of almost periodic functions with values in Banach spaces are extended to this case. Applications to fuzzy differential equations and to (fuzzy) dynamical systems are given.     

Monotone periodic orbits for torus homeomorphisms

Let be a homeomorphism of the torus isotopic to the identity and suppose that there exists a periodic orbit with a non-zero rotation vector . Then has a topologically monotone periodic orbit with the same rotation vector.

     

Almost periodic transformable difference systems

We analyse almost periodic homogeneous linear difference systems whose coefficient matrices belong to the so-called transformable groups. We introduce the notion of weakly transformable groups and we use this notion to obtain generalizations of known results about non-almost periodic solutions of considered systems.     

Almost periodic invariant tori for the NLS on the circle

In this paper we study the existence and linear stability of almost periodic solutions for a NLS equation on the circle with external parameters. Starting from the seminal result of Bourgain in [15] on the quintic NLS, we propose a novel approach allowing to prove in a unified framework the persistence of finite and infinite dimensional invariant tori, which are the support of the desired solutions. The persistence result is given through a rather abstract “counter-term theorem” à la Herman, directly in the original elliptic variables without passing to action-angle ones. Our framework allows us to find “many more” almost periodic solutions with respect to the existing literature and consider also non-translation invariant PDEs.     

Hypoelliptic vector fields and almost periodic motions on the torus Tn

Abstract: In this paper we prove that a hypoelliptic vector fields on the torus Tⁿ may be transformed by a smooth diffeomorphism of torus Tⁿ into a vector field with constant coefficients, which moreover satisfy a Diophantine condition. We also discuss the relation between the hypoelliptic vecttor fields and the almost periodic motions on the torus Tⁿ      

Almost periodic compactifications of group extensions

Let and be groups and let be an extension of by . Given a property of group compactifications, one can ask whether there exist compactifications and of N and K such that the universal -compactification of G is canonically isomorphic to an extension of by . We prove a theorem which gives necessary and sufficient conditions for this to occur for general properties and then apply this result to the almost periodic and weakly almost periodic compactifications of G.     

Almost periodic distributions in tube domains

The notion of an almost periodic (a.p.) distribution in a tube domain T_G in Cⁿ is introduced. Properties of such distributions are investigated; some of them are similar to the properties of ordinary a.p. functions on the axis. In these terms, the notion of an a.p. divisor in T_G is introduced, and conditions for the existence of the density of such a divisor in T_G′, { }, are found. Bibliography: 12 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 247, 1997, pp. 210–236. Translated by I. A. Fedortsova.     

Almost periodic divisors in a strip

Almost periodic divisors in a strip of the complex plane are studied and related to the divisors of holomorphic almost periodic functions in the strip.     

Almost periodic upper and lower solutions

The method of upper and lower solutions is a classical tool in the theory of periodic differential equations of the second order. We show that this method does not have a direct extension to almost periodic equations. To do this we construct equations of this type without almost periodic solutions but having two constants as ordered upper and lower solutions.