English

The "spectral" decomposition for one-dimensional maps

Dynamical Systems 2016-01-25 v1

Abstract

We construct the "spectral" decomposition of the sets Perfˉ\bar{Per\,f}, ω(f)=ω(x)\omega(f)=\cup\omega(x) and Ω(f)\Omega(f) for a continuous map ff of the interval to itself. Several corollaries are obtained; the main ones describe the generic properties of ff-invariant measures, the structure of the set Ω(f)Perfˉ\Omega(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.

Keywords

Cite

@article{arxiv.math/9201290,
  title  = {The "spectral" decomposition for one-dimensional maps},
  author = {Alexander M. Blokh},
  journal= {arXiv preprint arXiv:math/9201290},
  year   = {2016}
}