On minimality of nonautonomous dynamical systems
The minimality of a nonautonomous dynamical system given by a compact Hausdorff space X and a sequence of continuous selfmaps of X is studied. A sufficient condition for nonminimality of such a system is formulated. A special attention is paid to the particular case when X is a real compact interv...
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| Date: | 2004 |
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| Main Authors: | , , |
| Format: | Article |
| Language: | English |
| Published: |
Інститут математики НАН України
2004
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| Series: | Нелінійні коливання |
| Online Access: | http://dspace.nbuv.gov.ua/handle/123456789/176995 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | On minimality of nonautonomous dynamical systems / S.F. Kolyada, Ľ. Snoha, S.I. Trofimchuk // Нелінійні коливання. — 2004. — Т. 7, № 1. — С. 86-92. — Бібліогр.: 16 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| Summary: | The minimality of a nonautonomous dynamical system given by a compact Hausdorff space X and a
sequence of continuous selfmaps of X is studied. A sufficient condition for nonminimality of such a
system is formulated. A special attention is paid to the particular case when X is a real compact interval
I. A sequence of continuous selfmaps of I forming a minimal nonautonomous system may uniformly
converge. For instance, the limit may be any topologically transitive map. But if all the maps in the sequence
are surjective then the limit is necessarily monotone. An example is given when the limit is the identity. As
an application, in a simple way we construct a triangular map in the square I² with the property that every
point except of those in the leftmost fibre has an orbit whose ω-limit set coincides with the leftmost fibre. |
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