Combinatorics, geometry and attractors of quasi-quadratic maps
Dynamical Systems
2008-02-03 v1
Abstract
The Milnor problem on one-dimensional attractors is solved for S-unimodal maps with a non-degenerate critical point c. It provides us with a complete understanding of the possible limit behavior for Lebesgue almost every point. This theorem follows from a geometric study of the critical set of a "non-renormalizable" map. It is proven that the scaling factors characterizing the geometry of this set go down to 0 at least exponentially. This resolves the problem of the non-linearity control in small scales. The proofs strongly involve ideas from renormalization theory and holomorphic dynamics.
Cite
@article{arxiv.math/9212210,
title = {Combinatorics, geometry and attractors of quasi-quadratic maps},
author = {Mikhail Lyubich},
journal= {arXiv preprint arXiv:math/9212210},
year = {2008}
}