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We demonstrate the existence of a global attractor A with a Cantor set structure for the renormalization of critical circle mappings. The set A is invariant under a generalized renormalization transformation, whose action on A is conjugate to the two-sided shift.
We give an example of a totally disconnected set $E \subset {\mathbb R}^3$ which is not removable for quasiconformal homeomorphisms, i.e., there is a homeomorphism $f$ of ${\mathbb R}^3$ to itself which is quasiconformal off $E$, but not quasiconformal on all of ${\mathbb R}^3$. The set $E$ may be taken with Hausdorff dimension $2$. The construction also gives a non-removable set for locally biLipschitz homeomorphisms.
Brillouin zones were introduced by Brillouin in the thirties to describe quantum mechanical properties of crystals, that is, in a lattice in $\mathbb{R}^n$. They play an important role in solid-state physics. It was shown by Bieberbach that Brillouin zones tile the underlying space and that each zone has the same area. We generalize the notion of Brillouin Zones to apply to an arbitrary discrete set in a proper metric space, and show that analogs of Bieberbach's results hold in this context. We then use these ideas to discuss focusing of geodesics in orbifolds of constant curvature. In the particular case of the Riemann surfaces $\mathbb{H}^2/\Gamma (k)$ (k=2,3, or 5), we explicitly count the number of geodesics of length $t$ that connect the point $i$ to itself
Let $F$ be a quadratic rational map of the sphere which has two fixed Siegel disks with bounded type rotation numbers $\theta$ and $\nu$. Using a new degree 3 Blaschke product model for the dynamics of $F$ and an adaptation of complex a priori bounds for renormalization of critical circle maps, we prove that $F$ can be realized as the mating of two Siegel quadratic polynomials with the corresponding rotation numbers $\theta$ and $\nu$.
It has been known for some time that the topological entropy is a nondecreasing function of the parameter in the real quadratic family, which corresponds to the intuitive idea that more nonlinearity induces more complex dynamical behavior. Polynomial families of higher degree depend on several parameters, so that the very question of monotonicity needs to be reformulated. For instance, one can say the entropy is monotone in a multiparameter family if the isentropes, or sets of maps with the same topological entropy, are connected. Here we reduce the problem of the connectivity of the isentropes in the real cubic families to a weak form of the Fatou conjecture on generic hyperbolicity, which was proved to hold true by C. Heckman. We also develop some tools which may prove to be useful in the study of other parameterized families, in particular a general monotonicity result for stunted sawtooth maps: the stunted sawtooth family of a given shape can be understood as a simple family which realizes all the possible combinatorial structures one can expect with a map of this shape on the basis of kneading theory. Roughly speaking, our main result about real cubic families is that they are as monotone as the stunted sawtooth families with the same shapes because of Heckman's result (there are two posible shapes for cubic maps, depending on the behavior at infinity).
This manuscript is an introduction to the theory of holomorphic foliations on the complex projective plane. Historically the subject has emerged from the theory of ODEs in the complex domain and various attempts to solve Hilbert's 16th Problem, but with the introduction of complex algebraic geometry, foliation theory and dynamical systems, it has now become an interesting subject of its own.
Let $f(z) = z^2 + c$ be a quadratic polynomial, with c in the Mandelbrot set. Assume further that both fixed points of f are repelling, and that f is not renormalizable. Then we prove that the Julia set J of f is holomorphically removable in the sense that every homeomorphism of the complex plane to itself that is conformal off of J is in fact conformal on the entire complex plane. As a corollary, we deduce that the Mandelbrot Set is locally connected at such c.
A frequent problem in holomorphic dynamics is to prove local connectivity of Julia sets and of many points of the Mandelbrot set; local connectivity has many interesting implications. The intention of this paper is to present a new point of view for this problem: we introduce fibers of these sets, and the goal becomes to show that fibers are "trivial", i.e. they consist of single points. The idea is to show "shrinking of puzzle pieces" without using specific puzzles. This implies local connectivity at these points, but triviality of fibers is a somewhat stronger property than local connectivity. Local connectivity proofs in holomorphic dynamics often actually yield that fibers are trivial, and this extra knowledge is sometimes useful.
Since we believe that fibers may be useful in further situations, we discuss their properties for arbitrary compact connected and full sets in the complex plane. This allows to use them for connected filled-in Julia sets of polynomials, and we deduce for example that infinitely renormalizable polynomials of the form $z^d+c$ have the property that the impression of any dynamic ray at a rational angle is a single point. An appendix reviews known topological properties of compact, connected and full sets in the plane.
The definition of fibers grew out of a new brief proof that the Mandelbrot set is locally connected at every Misiurewicz point and at every point on the boundary of a hyperbolic component. This proof works also for "Multibrot sets", which are the higher degree cousins of the Mandelbrot set. These sets are discussed in a self-contained sequel (IMS Preprint 1998/13a). Finally, we relate triviality of fibers to tuning and renormalization in IMS Preprint 1998/13b.
We give new proofs that the Mandelbrot set is locally connected at every Misiurewicz point and at every point on the boundary of a hyperbolic component. The idea is to show "shrinking of puzzle pieces" without using specific puzzles. Instead, we introduce fibers of the Mandelbrot set and show that fibers of certain points are "trivial", i.e., they consist of single points. This implies local connectivity at these points.
Locally, triviality of fibers is strictly stronger than local connectivity. Local connectivity proofs in holomorphic dynamics often actually yield that fibers are trivial, and this extra knowledge is sometimes useful. We include the proof that local connectivity of the Mandelbrot set implies density of hyperbolicity in the space of quadratic polynomials.
We write our proofs more generally for Multibrot sets, which are the loci of connected Julia sets for polynomials of the form $z\mapsto z^d+c$.
Although this paper is a continuation of preprint 1998/12, it has been written so as to be independent of the discussion of fibers of general compact connected and full sets in $\mathbb{C}$ given there.
We continue the description of Mandelbrot and Multibrot sets and of Julia sets in terms of fibers which was begun in IMS preprints 1998/12 and 1998/13a. The question of local connectivity of these sets is discussed in terms of fibers and becomes the question of triviality of fibers. In this paper, the focus is on the behavior of fibers under renormalization and other surgery procedures. We show that triviality of fibers of Mandelbrot and Multibrot sets is preserved under tuning maps and other (partial) homeomorphisms. Similarly, we show for unicritical polynomials that triviality of fibers of Julia sets is preserved under renormalization and other surgery procedures, such as the Branner-Douady homeomorphisms. We conclude with various applications about quadratic polynomials and its parameter space: we identify embedded paths within the Mandelbrot set, and we show that Petersen's theorem about quadratic Julia sets with Siegel disks of bounded type generalizes from period one to arbitrary periods so that they all have trivial fibers and are thus locally connected.