## Institute for Mathematical Sciences

## Preprint ims91-15

** M. Lyubich & J. Milnor**
* The Fibonacci Unimodal Map.*

Abstract: This paper will study topological, geometrical and measure-theoretical properties of the real Fibonacci map. Our
goal was to figure out if this type of recurrence really gives
any pathological examples and to compare it with the infinitely
renormalizable patterns of recurrence studied by Sullivan. It
turns out that the situation can be understood completely and
is of quite regular nature. In particular, any Fibonacci map
(with negative Schwarzian and non-degenerate critical point)
has an absolutely continuous invariant measure (so, we deal
with a ``regular'' type of chaotic dynamics). It turns out also
that geometrical properties of the closure of the critical
orbit are quite different from those of the Feigenbaum map: its
Hausdorff dimension is equal to zero and its geometry is not
rigid but depends on one parameter.

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