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.
Keywords. Dynamical system, distributional chaos, basic sets.
MS 2000 Classification numbers. Primary 337D45, 37E10, 54H20.