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 $G$ be a non-elementary, finitely generated Kleinian group, $\Lambda(G)$ its limit set and $\Omega(G) = \overline {\mathbb C} \backslash \Lambda(G)$ its set of discontinuity. Let $\delta(G)$ be the critical exponent for the Poincarè series and let $\Lambda_c$ be the conical limit set of $G$. Suppose $\Omega_0$ is a simply connected component of $\Omega(G)$. We prove that

1. $\delta(G) = \dim(\Lambda_c)$
2. A simply connected component $\Omega$ is either a disk or $\dim(\partial \Omega)>1$
3. $ \Lambda(G)$ is either totally disconnected, a circle or has dimension $>1$
4. $G$ is geometrically infinite iff $\dim(\Lambda)=2$
5. If $G_n \to G$ algebraically then $\dim(\Lambda)\leq \liminf \dim(\Lambda_n)$
6. The Minkowski dimension of $\Lambda$ equals the Hausdorff dimension
7. If $\text{area}(\Lambda)=0$ then $\delta(G) =\dim(\Lambda(G))$

The proof also shows that $\dim(\Lambda(G)) >1$ iff the conical limit set has dimension $>1$ iff the Poincarè exponent of the group is $>1$. Furthermore, a simply connected component of $\Omega(G)$ either is a disk or has non-differentiable boundary in the the sense that the (inner) tangent points of $\partial \Omega$ have zero $1$-dimensional measure. almost every point (with respect to harmonic measure) is a twist point.

We construct a diffeomorphism of the two-dimensional torus which is isotopic to the identity and whose rotation set is not a polygon.

Given a $C^{1+\gamma}$ hyperbolic Cantor set $C$, we study the sequence $C_{n,x}$ of Cantor subsets which nest down toward a point $x$ in $C$. We show that $C_{n,x}$ is asymptotically equal to an ergodic Cantor set valued process. The values of this process, called limit sets, are indexed by a H\"older continuous set-valued function defined on D. Sullivan's dual Cantor set. We show the limit sets are themselves $C^{k+\gamma}, C^\infty$ or $C^\omega$ hyperbolic Cantor sets, with the highest degree of smoothness which occurs in the $C^{1+\gamma}$ conjugacy class of $C$. The proof of this leads to the following rigidity theorem: if two $C^{k+\gamma}, C^\infty$ or $C^\omega$ hyperbolic Cantor sets are $C^1$-conjugate, then the conjugacy (with a different extension) is in fact already $C^{k+\gamma}, C^\infty$ or $C^\omega$. Within one $C^{1+\gamma}$ conjugacy class, each smoothness class is a Banach manifold, which is acted on by the semigroup given by rescaling subintervals. Conjugacy classes nest down, and contained in the intersection of them all is a compact set which is the attractor for the semigroup: the collection of limit sets. Convergence is exponentially fast, in the $C^1$ norm.