## Institute for Mathematical Sciences

## Preprint ims92-18

** M. Lyubich**
* Combinatorics, Geometry and Attractors of Quasi-Quadratic Maps.*

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 $\omega(c)$
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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