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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
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.
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.
Motivated by the work of Douady, Ghys, Herman and Shishikura on Siegel quadratic polynomials, we study the one-dimensional slice of the cubic polynomials which have a fixed Siegel disk of rotation number $\theta$, with $\theta$ being a given irrational number of Brjuno type. Our main goal is to prove that when $\theta$ is of bounded type, the boundary of the Siegel disk is a quasicircle which contains one or both critical points of the cubic polynomial. We also prove that the locus of all cubics with both critical points on the boundary of their Siegel disk is a Jordan curve, which is in some sense parametrized by the angle between the two critical points. A main tool in the bounded type case is a related space of degree 5 Blaschke products which serve as models for our cubics. Along the way, we prove several results about the connectedness locus of these cubic polynomials.
We study the dynamics of polynomial mappings $f:{\bf C}^k\to{\bf C}^k$ of degree $d\ge2$ that extend continuously to projective space ${\bf P}^k$. Our approach is to study the dynamics near the hyperplane at infinity and then making a descent to $K$ --- the set of points with bounded orbits --- via external rays.
The Poisson boundary of a group $G$ with a probability measure $\mu$ on it is the space of ergodic components of the time shift in the path space of the associated random walk. Via a generalization of the classical Poisson formula it gives an integral representation of bounded $\mu$-harmonic functions on $G$. In this paper we develop a new method of identifying the Poisson boundary based on entropy estimates for conditional random walks. It leads to simple purely geometric criteria of boundary maximality which bear hyperbolic nature and allow us to identify the Poisson boundary with natural topological boundaries for several classes of groups: word hyperbolic groups and discontinuous groups of isometries of Gromov hyperbolic spaces, groups with infinitely many ends, cocompact lattices in Cartan--Hadamard manifolds, discrete subgroups of semi-simple Lie groups, polycyclic groups, some wreath and semi-direct products including Baumslag--Solitar groups.
Let $f$ be a quadratic polynomial which has an irrationally indifferent fixed point $\alpha$. Let $z$ be a biaccessible point in the Julia set of $f$. Then:
- In the Siegel case, the orbit of $z$ must eventually hit the critical point of $f$.
- In the Cremer case, the orbit of $z$ must eventually hit the fixed point $\alpha$.
Siegel polynomials with biaccessible critical point certainly exist, but in the Cremer case it is possible that biaccessible points can never exist.
As a corollary, we conclude that the set of biaccessible points in the Julia set of a Siegel or Cremer quadratic polynomial has Brolin measure zero.
We prove that the only possible biaccessible points in the Julia set of a Cremer quadratic polynomial are the Cremer fixed point and its preimages. This gives a partial answer to a question posed by C. McMullen on whether such a Julia set can contain any biaccessible point at all.
Let $f:z \mapsto z^2+c$ be a quadratic polynomial whose Julia set $J$ is locally-connected. We prove that the Brolin measure of the set of biaccessible points in $J$ is zero except when $f(z)=z^2-2$ is the Chebyshev quadratic polynomial for which the corresponding measure is one.
We show that the Feigenbaum-Cvitanović equation can be interpreted as a linearizing equation, and the domain of analyticity of the Feigenbaum fixed point of renormalization as a basin of attraction. There is a natural decomposition of this basin which enables to recover a result of local connectivity by Jiang and Hu for the Feigenbaum Julia set.