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This paper proves a theorem about bounding orbits of a time dependent dynamical system. The maps that are involved are examples in convex dynamics, by which we mean the dynamics of piecewise isometries where the pieces are convex. The theorem came to the attention of the authors in connection with the problem of digital halftoning. $\textit{Digital halftoning}$ is a family of printing technologies for getting full color images from only a few different colors deposited at dots all of the same size. The simplest version consist in obtaining grey scale images from only black and white dots. A corollary of the theorem is that for $\textit{error diffusion}$, one of the methods of digital halftoning, averages of colors of the printed dots converge to averages of the colors taken from the same dots of the actual images. Digital printing is a special case of a much wider class of scheduling problems to which the theorem applies. Convex dynamics has roots in classical areas of mathematics such as symbolic dynamics, Diophantine approximation, and the theory of uniform distributions.
An exposition of the 1918 paper of Lattès and its modern formulations and applications.
Let $(M,J,\Omega)$ be a polarized complex manifold of Kähler type. Let $G$ be the maximal compact subgroup of the automorphism group of $(M,J)$. On the space of Kähler metrics that are invariant under $G$ and represent the cohomology class $\Omega$, we define a flow equation whose critical points are extremal metrics, those that minimize the square of the $L^2$-norm of the scalar curvature. We prove that the dynamical system in this space of metrics defined by the said flow does not have periodic orbits, and that its only fixed points, or extremal solitons, are extremal metrics. We prove local time existence of the flow, and conclude that if the lifespan of the solution is finite, then the supremum of the norm of its curvature tensor must blow-up as time approaches it. We end up with some conjectures concerning the plausible existence and convergence of global solutions under suitable geometric conditions.
The object of this paper is to prove some general results about rational idempotents for a finite group $G$ and deduce from them geometric information about the components that appear in the decomposition of the Jacobian variety of a curve with $G-$action.
We give an algorithm to find explicit primitive rational idempotents for any $G$, as well as for rational projectors invariant under any given subgroup. These explicit constructions allow geometric descriptions of the factors appearing in the decomposition of a Jacobian with group action: from them we deduce the decomposition of any Prym or Jacobian variety of an intermediate cover, in the case of a Jacobian with $G-$action. In particular, we give a necessary and sufficient condition for a Prym variety of an intermediate cover to be such a factor.
The goal of this technical note is to show that the geometry of generalized parabolic towers cannot be essentially bounded. It fills a gap in author's paper "Combinatorics, geomerty and attractors of quasi-quadratic maps", Annals of Math., 1992.
David maps are generalizations of classical planar quasiconformal maps for which the dilatation is allowed to tend to infinity in a controlled fashion. In this note we examine how these maps distort Hausdorff dimension. We show
- Given $\alpha$ and $\beta$ in $[0,2]$, there exists a David map $\varphi:\mathbb {C} \to \mathbb {C}$ and a compact set $\Lambda$ such that $\operatorname{dim}_{\operatorname{H}} \Lambda =\alpha$ and $\operatorname{dim}_{\operatorname{H}} \varphi(\Lambda)=\beta$.
- There exists a David map $\varphi:\mathbb {C} \to \mathbb {C}$ such that the Jordan curve $\Gamma=\varphi ({\mathbb S}^1)$ satisfies $\operatorname{dim}_{\operatorname{H}} \Gamma=2$.
One should contrast the first statement with the fact that quasiconformal maps preserve sets of Hausdorff dimension $0$ and $2$. The second statement provides an example of a Jordan curve with Hausdorff dimension $2$ which is (quasi)conformally removable.
A homeomorphism of a compact metric space is tight provided every non-degenerate compact connected (not necessarily invariant) subset carries positive entropy. It is shown that every $C^{1+\alpha}$ diffeomorphism of a closed surface factors to a tight homeomorphism of a generalized cactoid (roughly, a surface with nodes) by a semi-conjugacy whose fibers carry zero entropy.
In this work, under a mild assumption, we give the classification of the complete polynomial vector fields in two variables up to algebraic automorphisms of $\mathbb{C}^2$. The general problem is also reduced to the study of the combinatorics of certain resolutions of singularities. Whereas we deal with $\mathbb{C}$-complete vector fields, our results also apply to $\mathbb{R}$-complete ones thanks to a theorem of Forstneric [Fo].
This paper investigates the set of angles of the parameter rays which land on the real slice [-2, 1/4] of the Mandelbrot set. We prove that this set has zero length but Hausdorff dimension 1. We obtain the corresponding results for the tuned images of the real slice. Applications of these estimates in the study of critically non-recurrent real quadratics as well as biaccessible points of quadratic Julia sets are given.
Global results are proved about the way in which Boyland's forcing partial order organizes a set of braid types: those of periodic orbits of Smale's horseshoe map for which the associated train track is a star. This is a special case of a conjecture introduced in [1], which claims that forcing organizes all horseshoe braid types into linearly ordered families which are, in turn, parametrized by homoclinic orbits to the fixed point of code 0.