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PREPRINTS IN THIS SERIES, IN PDF FORMAT.
* Starred papers have appeared in the journal cited.
We study the the tangent family $\mathcal{F} = \{\lambda \tan z, \lambda \in \mathbb{C} - \{0\}\}$ and give a complete classification of their stable behavior. We also characterize the the hyperbolic components and give a combinatorial description their deployment in the parameter plane.
Consider a planar Brownian motion run for finite time. The frontier or "outer boundary" of the path is the boundary of the unbounded component of the complement. Burdzy (1989) showed that the frontier has infinite length. We improve this by showing that the Hausdorff dimension of the frontier is strictly greater than 1. (It has been conjectured that the Brownian frontier has dimension $4/3$, but this is still open). The proof uses Jones's Traveling Salesman Theorem and a self-similar tiling of the plane by fractal tiles known as Gosper Islands.
Let $M$ be a compact 3-manifold whose interior admits a complete hyperbolic structure. We let $\Lambda(M)$ be the supremum of $\lambda_0(N)$ where $N$ varies over all hyperbolic 3-manifolds homeomorphic to the interior of $N$. Similarly, we let $D(M)$ be the infimum of the Hausdorff dimensions of limit sets of Kleinian groups whose quotients are homeomorphic to the interior of $M$. We observe that $\Lambda(M)=D(M)(2-D(M))$ if $M$ is not handlebody or a thickened torus. We characterize exactly when $\Lambda(M)=1$ and $D(M)=1$ in terms of the characteristic submanifold of the incompressible core of $M$.
Varchenko conjectured that, under certain genericity conditions, the number of critical points of a product $\phi$ of powers of linear functions on $\mathbb {C}^n$ should be given by the Euler characteristic of the complement of the divisor of $\phi$ (i.e., a union of hyperplanes). In this note two independent proofs are given of a direct generalization of Varchenko's conjecture to the case of a generalized meromorphic function on an algebraic manifold whose divisor can be any (generally singular) hypersurface. The first proof uses characteristic classes and a formula of Gauss--Bonnet type for affine algebraic varieties. The second proof uses Morse theory.
We use the methods developed with M. Lyubich for proving complex bounds for real quadratics to extend E. De Faria's complex a priori bounds to all critical circle maps with an irrational rotation number. The contracting property for renormalizations of critical circle maps follows. In the Appendix we give an application of the complex bounds for proving local connectivity of some Julia sets.
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.