Combinatorics, geometry and attractors of quasi-quadratic maps

Details Combinatorics, geometry and attractors of quasi-quadratic maps Combinatorics, geometry and attractors of quasi-quadratic maps

1992-12-06

arxiv.org/abs/math/9212210v1

Mathematics

Dynamical Systems

Scientific paper

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 $\omega(c)$ 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.

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