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Combinatorics, geometry and attractors of quasi-quadratic maps

Mikhail Lyubich
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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 $ω(c)$ 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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Mikhail Lyubich. 1992-12-06. Combinatorics, geometry and attractors of quasi-quadratic maps. https://arxiv.org/abs/math/9212210

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