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| - Schweizer and Smital [Tran. Amer. Math. Soc. 344 (1994), 737--754] introduced the notion of distributional chaos for continuous maps of the interval. In this paper we show that for the continuous mappings of the circle the results are very similar, up to natural modifications. Thus any such mapping has a finite spectrum, which is generated by the map restricted to a finite collection of basic sets, and any scrambled set in the sense of Li and Yorke has a decomposition into three subsets (on the interval into two subsets) such that the distribution function generated on any such subset is lower bounded by a distribution function from the spectrum. While the results are similar, the original argument is not applicable directly and needs essential modifications. Thus, e.g., we had first to develop the theory of basic sets on the circle.
- Schweizer and Smital [Tran. Amer. Math. Soc. 344 (1994), 737--754] introduced the notion of distributional chaos for continuous maps of the interval. In this paper we show that for the continuous mappings of the circle the results are very similar, up to natural modifications. Thus any such mapping has a finite spectrum, which is generated by the map restricted to a finite collection of basic sets, and any scrambled set in the sense of Li and Yorke has a decomposition into three subsets (on the interval into two subsets) such that the distribution function generated on any such subset is lower bounded by a distribution function from the spectrum. While the results are similar, the original argument is not applicable directly and needs essential modifications. Thus, e.g., we had first to develop the theory of basic sets on the circle. (en)
- Schweizer a Smítal (porovnej s Trans. Amer. Math. Soc. 344 (1944) 737-754) zavedli pojem distribučního chaosu pro spojitou funkci intervalu. V této práci ukážeme, že obdobné výsledky, až na drobné ale přirozené odlišnosti, lze obdržet také pro případ spojitého zobrazení ružnice. Tedy každé takové zobrazení má konečné spektrum, které je generováno zúžením zobrazení na konečný počet bázických množin, a že každá chaotická množina v Li-Yorkově smyslu má rozklad na tři podmnožiny (v případě intervalového zobrazení jsou dvě) takové, že distribuční funkce generovaná body z těchto podmnožin je zdola ohraničená distribuční funkcí ze spektra. Ačkoli jsou tyto výsledky shodné s intervalovými, původní argumentaci nelze použít přímo, ale je nutno provést značné modifikace. (cs)
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| Title
| - Distributional chaos and spectral decomposition of dynamical systems on the circle
- Distributional chaos and spectral decomposition of dynamical systems on the circle (en)
- Distribuční chaos a spektrální rozklad dynamického systému na kružnici (cs)
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| skos:prefLabel
| - Distributional chaos and spectral decomposition of dynamical systems on the circle
- Distributional chaos and spectral decomposition of dynamical systems on the circle (en)
- Distribuční chaos a spektrální rozklad dynamického systému na kružnici (cs)
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| skos:notation
| - RIV/47813059:19610/04:00011712!RIV/2005/GA0/196105/N
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| http://linked.open.../vavai/riv/strany
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| http://linked.open...avai/riv/aktivita
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| http://linked.open...avai/riv/aktivity
| - P(GA201/00/0859), Z(MSM 192400002)
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| http://linked.open...iv/cisloPeriodika
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| http://linked.open...vai/riv/dodaniDat
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| http://linked.open...aciTvurceVysledku
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| http://linked.open.../riv/druhVysledku
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| http://linked.open...iv/duvernostUdaju
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| http://linked.open...titaPredkladatele
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| http://linked.open...dnocenehoVysledku
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| http://linked.open...ai/riv/idVysledku
| - RIV/47813059:19610/04:00011712
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| http://linked.open...riv/jazykVysledku
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| http://linked.open.../riv/klicovaSlova
| - Dynamical system;distributional chaos;basic sets;spectral decomposition (en)
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| http://linked.open.../riv/klicoveSlovo
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| http://linked.open...odStatuVydavatele
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| http://linked.open...ontrolniKodProRIV
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| http://linked.open...i/riv/nazevZdroje
| - Topology and its Applications
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| http://linked.open...in/vavai/riv/obor
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| http://linked.open...ichTvurcuVysledku
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| http://linked.open...cetTvurcuVysledku
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| http://linked.open...vavai/riv/projekt
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| http://linked.open...UplatneniVysledku
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| http://linked.open...v/svazekPeriodika
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| http://linked.open...iv/tvurceVysledku
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| http://linked.open...n/vavai/riv/zamer
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| issn
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| number of pages
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| http://localhost/t...ganizacniJednotka
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