Institute for Mathematical Sciences

Preprint ims91-14

A. M. Blokh
The "Spectral" Decomposition for One-Dimensional Maps

Abstract: We construct the ``spectral'' decomposition of the sets $\overline{Per\,f}$, $\omega(f)=\cup\omega(x)$ and $\Omega(f)$ 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 $\Omega(f)\setminus \overline{Per\,f}$ 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.
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