• A
  • A
  • A
  • АБB
  • АБB
  • АБB
  • А
  • А
  • А
  • А
  • А
Обычная версия сайта
Найдены 3 публикации
Сортировка:
по названию
по году
Статья
Buff X., Goncharuk Nataliya. Journal of Modern Dynamics. 2015. Vol. 9. P. 169-190.

We investigate the notion of complex rotation number which was introduced by V.I.Arnold in 1978. Let f: R/Z ->  R/Z be a (real) analytic orientation preserving circle diffeomorphism and let omega in C/Z be a parameter with positive imaginary part. Construct a complex torus by glueing the two boundary components of the annulus { z in C/Z : 0< Im(z) < Im(omega)} via the map f+omega. This complex torus is isomorphic to C/(Z+ tau Z) for some appropriate tau in C/Z.   According to V.Moldavskis, if the ordinary rotation number rot(f+omega0) is Diophantine and if omega tends to omega0 non tangentially to the real axis, then tau tends to rot(f+omega0). We show that the Diophatine and non tangential assumptions are unnecessary: if rot(f+omega0) is irrational then tau tends to rot(f+omega0) as omega tends to omega0.  This, together with results of N. Goncharuk [4], motivates us to introduce a new fractal set (``bubbles'') given by the limit values of tau as omega tends to the real axis. For the rational values of rot (f+omega0), these limits do not necessarily coincide with rot(f+omega0) and form a countable number of analytic loops in the upper half-plane.

Добавлено: 10 октября 2013
Статья
Skripchenko A., Dynnikov I. Journal of Modern Dynamics. 2017. Vol. 11. P. 219-248.
Добавлено: 20 апреля 2018
Статья
Glutsyuk A., Kudryashov Y. Journal of Modern Dynamics. 2012. Vol. 6. No. 3. P. 287-326.

The article is devoted to a particular case of Ivrǐ's conjecture on periodic orbits of billiards. The general conjecture states that the set of periodic orbits of the billiard in a domain with smooth boundary in the Euclidean space has measure zero. In this article we prove that for any domain with piecewise C 4-smooth boundary in the plane the set of quadrilateral trajectories of the corresponding billiard has measure zero.

Добавлено: 5 февраля 2013