## Institute for Mathematical Sciences

## Preprint ims93-12

** Mikhail Lyubich**
* Teichmuller space of Fibonacci maps*

Abstract: According to Sullivan, a space ${\cal E}$ of unimodal maps with the same combinatorics (modulo smooth conjugacy) should be
treated as an infinitely-dimensional Teichm\"{u}ller space.
This is a basic idea in Sullivan's approach to the
Renormalization Conjecture. One of its principle ingredients
is to supply ${\cal E}$ with the Teichm\"{u}ller metric. To
have such a metric one has to know, first of all, that all maps
of ${\cal E}$ are quasi-symmetrically conjugate. This was
proved [Ji] and [JS] for some classes of non-renormalizable
maps (when the critical point is not too recurrent). Here we
consider a space of non-renormalizable unimodal maps with in
a sense fastest possible recurrence of the critical point
(called Fibonacci). Our goal is to supply this space with the
Teichm\"{u}ller metric.

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