Teichm\"uller space of Fibonacci maps
Abstract
According to Sullivan, a space 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 with the Teichm\"{u}ller metric. To have such a metric one has to know, first of all, that all maps of 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.
Cite
@article{arxiv.math/9311213,
title = {Teichm\"uller space of Fibonacci maps},
author = {Mikhail Lyubich},
journal= {arXiv preprint arXiv:math/9311213},
year = {2016}
}