The "spectral" decomposition for one-dimensional maps

Abstract

We construct the "spectral" decomposition of the sets $\bar{Per\,f}$, $ω(f)=\cupω(x)$ and $Ω(f)$ for a continuous map $f$ of the interval to itself. Several corollaries are obtained; the main ones describe the generic properties of $f$-invariant measures, the structure of the set $Ω(f)\setminus \bar{Per\,f}$ and the generic limit behavior of an orbit for maps without wandering intervals. The "spectral" decomposition for piecewise-monotone maps is deduced from the Decomposition Theorem. Finally we explain how to extend the results of the present paper for a continuous map of a one-dimensional branched manifold into itself.