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PREPRINTS IN THIS SERIES, IN PDF FORMAT.
* Starred papers have appeared in the journal cited.
We show that in any family of stunted sawtooth maps, the set of maps whose set of periods is the set of all powers of 2 has no interior point, i.e., the combinatorial description of the boundary of chaos coincides with the topological description. We also show that, under mild assumptions, smooth multimodal maps whose set of periods is the set of all powers of 2 are infinitely renormalizable.
This is a continuation of the series of notes on the dynamics of quadratic polynomials. We show the following
Rigidity Theorem: Any combinatorial class contains at most one quadratic polynomial satisfying the secondary limbs condition with a-priori bounds.
As a corollary, such maps are combinatorially and topologically rigid, and as a consequence, the Mandelbrot set is locally connected at the corresponding parameter values.
We extend Sullivan's complex a priori bounds to real quadratic polynomials with essentially bounded combinatorics. Combined with the previous results of the first author, this yields complex bounds for all real quadratics. Local connectivity of the corresponding Julia sets follows.
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.
In this paper we shall show that there exists a polynomial unimodal map $f: [0,1] \mapsto [0,1]$ which is
1) non-renormalizable(therefore for each x from a residual set, $\omega(x)$ is equal to an interval)
2) for which $\omega(c)$ is a Cantor set
3) for which $\omega(x)=\omega(c)$ for Lebesgue almost all x.
So the topological and the metric attractor of such a map do not coincide. This gives the answer to a question posed by Milnor.
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.
We show that the conjugacy class of an eventually expanding continuous piecewise affine interval map is contained in a smooth codimension 1 submanifold of parameter space. In particular conjugacy classes have empty interior. This is based on a study of the relation between induced Markov maps and ergodic theoretical behavior.
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
- $\delta(G) = \dim(\Lambda_c)$
- A simply connected component $\Omega$ is either a disk or $\dim(\partial \Omega)>1$
- $ \Lambda(G)$ is either totally disconnected, a circle or has dimension $>1$
- $G$ is geometrically infinite iff $\dim(\Lambda)=2$
- If $G_n \to G$ algebraically then $\dim(\Lambda)\leq \liminf \dim(\Lambda_n)$
- The Minkowski dimension of $\Lambda$ equals the Hausdorff dimension
- 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.
Let $H^d$ be the set of all rational maps of degree $d\ge 2$ on the Riemann sphere which are expanding on Julia set. We prove that if $f\in H^d$ and all or all but one critical points (or values) are in the immediate basin of attraction to an attracting fixed point then there exists a polynomial in the component $H(f)$ of $H^d$ containing $f$. If all critical points are in the immediate basin of attraction to an attracting fixed point or parabolic fixed point then $f$ restricted to Julia set is conjugate to the shift on the one-sided shift space of $d$ symbols. We give exotic examples of maps of an arbitrary degree $d$ with a non-simply connected, completely invariant basin of attraction and arbitrary number $k\ge 2$ of critical points in the basin. For such a map $f\in H^d$ with $k < d$ there is no polynomial in $H(f)$. Finally we describe a computer experiment joining an exotic example to a Newton's method (for a polynomial) rational function with a 1-parameter family of rational maps.
We construct a diffeomorphism of the two-dimensional torus which is isotopic to the identity and whose rotation set is not a polygon.