ALMOST PERIODIC MOTION IN COMPLETE SPACE ANANT KUMAR

It is interesting that in a complete space an almost periodic motion is periodic if the null set ϕ is closed and if ϕ is not closed then every point of its trajectory is a limit point.

periodic if the null set is closed and if is not closed then every point of its trajectory is a limit point.
KEY WORDS AND PHRASES.Compact, periodic and almost periodic motions.
We studied "Poisson stable distal dynamical systems" in Ill where the word "compact" is missing from the statement and proof of theorem 2.7.It should be stated as theorem 3.1(below).Now by means of this theorem we shall establish an interesting relation (theorem 3.2) between almost periodic and periodic motions in a complete space X.

DEFINITIONS AND NOTATIONS.
We shall follow definitions and notations of [i].
3.1 THEOREM.Compact almost periodic motion is Poisson stable and distal.
3.2 THEOREM.In a complete space X an almost periodic motion is periodic if the null set is closed in X and if is not closed then y(x) which is perfect and x compact set.
PROOF.An almost periodic motion (x,t) is recurrent [2, theo 8.02 P.384] and if a recurrent motion is situated in a complete space then cly(x) is compact minimal [2,   theo 7.07 P.377].Therefore the motion (x,t) is compact.Hence (x,t) is compact almost periodic motion therefore it is Poisson stable and distal (3.1 above).Thus y(x) is closed and perfect set [, theo 2.1].Now by [theo.VI.3 of 3, P.87], if x is not periodic then cl(cly(x) y(x)) cly(x) x Which is impossible, as xey(x) and in case of a compact motion is also non-empty x [3, theo.II.8 P.20].Therefore x is periodic.But if is not closed then cl x) //.

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).
We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: x3.3 COROLLARY.If in theorem 3.2,T=R (the set of reals) then y(x) is connected.

First
Round of Reviews March 1, 2009