In this paper we shall show that there exists $l_0$ such that for each even integer $l \geq l_0$ there exists $c_1 \in \mathbb{R}$ for which the Julia set of $z \mapsto z^l + c_1$ has positive Lebesgue measure. This solves an old problem.

Editor's note: In 1997, it was shown by Xavier Buff that there was a serious flaw in the argument, leaving a gap in the proof. Currently (1999), the question of polynomials with a positive measure Julia sets remains open.

Let $H: \mathbb{C}^2 \to \mathbb{C}^2$ be the Hénon mapping given by $$ \begin{bmatrix}x\\y\end{bmatrix} \mapsto \begin{bmatrix}p(x) - ay\\x\end{bmatrix}. $$ The key invariant subsets are $K_\pm$, the sets of points with bounded forward images, $J_\pm = \partial K_\pm$ their boundaries, $J = J_+ \cap J_-$, and $K = K_+ \cap K_-$. In this paper we identify the topological structure of these sets when $p$ is hyperbolic and $|a|$ is sufficiently small, \ie, when $H$ is a small perturbation of the polynomial $p$. The description involves projective and inductive limits of objects defined in terms of $p$ alone.

This is an outline of work in progress. We study the conjecture that the topological entropy of a real cubic map depends "monotonely" on its parameters, in the sense that each locus of constant entropy in parameter space is a connected set. This material will be presented in more detail in a later paper.

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).