On Dynamics of Total Expanding Mappings on R
Author(s):
Sergey Andreevich Brygin
Saint-Petersburg State University,
student
Universitetskiy Prospect,28
198504, Saint-Petersburg, Russia
Phone +79117008394,
sergeybrygen@mail.ru
Alexander Alekseevich Florinskiy
Saint-Petersburg State University,
Associated Professor,
PhD in Physics and Mathematics
Universitetskiy Prospect,28
198504, Saint-Petersburg, Russia
florinskiy.a@gmail.com
Abstract:
In this paper we show that there exists a
smooth transformation of real line, such that the sequence
of images of any nonempty open set under iterations of this
transformation has real line as its lower limit.
It is also proved that for such a transformation
there always exists a compact set having a dense
orbit in the space of all the compact subsets of real
line with Hausdorff metric. Some properties of such compact sets are considered.
References:
- H. W. Broer, F. Dumortier, S. J. van Strien, F. Takens. Structures in dynamics. Finite dimensional deterministic studies. Elivier Science Publishers, 1991, 336 p
- Brin M., Stuck G. Introduction to Dynamical Systems. Cambridge, Cambridge University Press (Virtual Publishing), 2003, 240 p
- R. M. Crownover. Introduction to fractals and Chaos. Boston, Jones and Bartlett, 1995, 306 p
- Y. G. Borisovich, B. D. Gelman, A. D. Myshkis, V. V. Obukhovskiy. Vvedenie v teoriyu mnogoznachnykh otobrazheniy i differentsial'nykh vklyucheniy [Introduction to the theory of multivalued mappings and differential inclusions]. M. : Editorial URSS, 2011, 216 р
- S. Y. Pilyugin. Limit sets of trajectories of regions in dynamical systems, Functional Analysis and Its Applications, July-September, 1989, Volume 23, Issue 3, pp. 242-243
- J. Oxtoby. Measure and Сategory. Springer-Verlag, Berlin, 1971
- B. M. Makarov, M. G. Goluzina, A. A. Lodkin, A. N. Podkorytov. Problemes d'analise reele. Cassini, Paris, 2010, 593 p
- N. M. Zobin, S. G. Crane. Matemamicheskiy analiz gladkih funktsiy [Mathematical analysis of smooth functions]. Voronezh, VSU, 1978, 144p
