## Institute for Mathematical Sciences

## Preprint ims93-9

** Mikhail Lyubich**
* Geometry of Quadratic Polynomials: Moduli, Rigidity and Local Connectivity.*

Abstract: A key problem in holomorphic dynamics is to classify complex quadratics $z\mapsto z^2+c$ up to topological conjugacy. The
Rigidity Conjecture would assert that any non-hyperbolic
polynomial is topologically rigid, that is, not topologically
conjugate to any other polynomial. This would imply density of
hyperbolic polynomials in the complex quadratic family (Compare
Fatou [F, p. 73]). A stronger conjecture usually abbreviated as
MLC would assert that the Mandelbrot set is locally connected.
A while ago MLC was proven for quasi-hyperbolic points by
Douady and Hubbard, and for boundaries of hyperbolic components
by Yoccoz. More recently Yoccoz proved MLC for all at most
finitely renormalizable parameter values. One of our goals is
to prove MLC for some infinitely renormalizable parameter
values. Loosely speaking, we need all renormalizations to have
bounded combinatorial rotation number (assumption C1) and
sufficiently high combinatorial type (assumption C2).
For real quadratic polynomials of bounded combinatorial type
the complex a priori bounds were obtained by Sullivan. Our
result complements the Sullivan's result in the unbounded case.
Moreover, it gives a background for Sullivan's renormalization
theory for some bounded type polynomials outside the real line
where the problem of a priori bounds was not handled before for
any single polynomial. An important consequence of a priori
bounds is absence of invariant measurable line fields on the
Julia set (McMullen) which is equivalent to quasi-conformal
(qc) rigidity. To prove stronger topological rigidity we
construct a qc conjugacy between any two topologically
conjugate polynomials (Theorem III). We do this by means of a
pull-back argument, based on the linear growth of moduli and a
priori bounds. Actually the argument gives the stronger
combinatorial rigidity which implies MLC.
We complete the paper with an application to the real
quadratic family. Here we can give a precise dichotomy (Theorem
IV): on each renormalization level we either observe a big
modulus, or essentially bounded geometry. This allows us to
combine the above considerations with Sullivan's argument for
bounded geometry case, and to obtain a new proof of the
rigidity conjecture on the real line (compare McMullen and
Swiatek).

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