Distributional chaos and spectral decomposition on the circle

Schweizer and Smı́tal [Trans. 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 similar results, up to natural modifications, are valid for the continuous mappings of the circle. Thus any such map 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 bounded from below by a distribution function from the spectrum. While the results are similar, the original argument is not applicable directly and needs essential modifications.