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In this paper, we will study Newton's method for solving two simultaneous quadratic equations in two variables. Presumably, there is no need to motivate a study of Newton's method, in one or several variables. The algorithm is of immense importance, and understanding its behavior is of obvious interest. It is perhaps harder to motivate the case of two simultaneous quadratic equations in two variables, but this is the simplest non-degenerate case.
It is well known that the Hausdorff dimension of the invariant set $\Lambda_t$ of an iterated function system ${\mathcal F}_t$ on $\mathbb{R}^n$ depending smoothly on a parameter $t$ does not vary continuously. In fact, it has been shown recently that in general it varies lower-semi-continuously. For a specific family of systems we investigate numerically the conjecture that discontinuities in the dimension only arise when in some iterate of the iterated function system two (or more) of its branches coincide. This happens in a set of co-dimension one, but which is dense. All the other points are conjectured to be points of continuity.
We discuss the dynamics of exponential maps $z\mapsto \lambda e^z$ from the point of view of dynamic rays, which have been an important tool for the study of polynomial maps. We prove existence of dynamic rays with bounded combinatorics and show that they contain all points which ``escape to infinity'' in a certain way. We then discuss landing properties of dynamic rays and show that in many important cases, repelling and parabolic periodic points are landing points of periodic dynamic rays. For the case of postsingularly finite exponential maps, this needs the use of spider theory.
We investigate closures of orbits for the action of the group of diagonal matrices acting on $SL(n,R)/SL(n,Z)$, where $n \geq 3$. It has been conjectured by Margulis that possible orbit-closures for this action are very restricted. Lending support to this conjecture, we show that any orbit-closure containing a compact orbit is homogeneous. Moreover if $n$ is prime then any orbit whose closure contains a compact orbit is either compact itself or dense. This implies a number-theoretic result generalizing an isolation theorem of Cassels and Swinnerton-Dyer for products of linear forms. We also obtain similar results for other lattices instead of $SL(n,Z)$, under a suitable irreducibility hypothesis.
Extending results of a number of authors, we prove that if $U$ is the unipotent radical of a solvable epimorphic subgroup of an algebraic group $G$, then the action of $U$ on $G/\Gamma$ is uniquely ergodic for every cocompact lattice $\Gamma$ in $G$. This gives examples of uniquely ergodic and minimal two-dimensional flows on homogeneous spaces of arbitrarily high dimension. Our main tools are Ratner classification of ergodic invariant measures for the action of a unipotent subgroup on a homogeneous space, and a simple lemma (the 'Cone Lemma') about representations of epimorphic subgroups. (revised version of July 1999)
We show that for an inclusion $F < G < L$ of real algebraic groups such that $F$ is epimorphic in $G$, any closed $F$-invariant subset of $L/\Lambda$ is $G$-invariant, where $\Lambda$ is a latice in $G$. This is a topological analogue of a result due to S. Mozes that any finite $F$-invariant measure on $L/\Lambda$ is $G$-invariant.
The key ingredient in establishing this result is the study of the limiting distributions of certain translates of a homogeneous measure. We show that if in addition $G$ is generated by unipotent elements then there exists $a\in F$ such that the following holds: Let $U\subset F$ be the subgroup generated by all unipotent elements of $F$, $x\in L/\Lambda$, and $\lambda$ and $\mu$ denote the Haar probability measures on the homogeneous spaces $\overline{Ux}$ and $\overline{Gx}$, respectively (cf.~Ratner's theorem). Then $a^n\lambda\to\mu$ weakly as $n\to\infty$.
We also give an algebraic characterization of algebraic subgroups $F<{SL}_n(\mathbb{R})$ for which all orbit closures are finite volume almost homogeneous spaces, namely ${\textit iff}$ the smallest observable subgroup of ${SL}_n(\mathbb{R})$ containing $F$ has no nontrivial characters defined over $\mathbb{R}$.
This short note serves as a joint introduction to the papers "On Actions of Epimorphic Subgroups on Homogeneous Spaces" by Nimesh Shah and Barak Weiss (Stony Brook IMS preprint 1999/7b) and "Unique Ergodicity on Compact Homogeneous Spaces" by Barak Weiss. For the benefit of the readers who are not experts in the theory of subgroup actions on homogeneous spaces I have prefaced the papers with some general remarks explaining and motivating our results, and the connection between them. The remarks are organized as a comparison between facts which had been previously known about the action of the geodesic and horocycle flow on finite-volume Riemann surfaces -- the simplest nontrivial example that falls into our framework -- and our results on subgroup actions on homogeneous spaces.
Given $C^2$ infinitely renormalizable unimodal maps $f$ and $g$ with a quadratic critical point and the same bounded combinatorial type, we prove that they are $C^{1+\alpha}$ conjugate along the closure of the corresponding forward orbits of the critical points, for some $\alpha>0$.
We show that the absolute Galois group acts faithfully on the set of Hubbard trees. Hubbard trees are finite planar trees, equipped with self-maps, which classify postcritically finite polynomials as holomorphic dynamical systems on the complex plane. We establish an explicit relationship between certain Hubbard trees and the trees known as "dessins d'enfant" introduced by Grothendieck.
Let $f(z)=e^{2i\pi\theta} z+z^2$, where $\theta$ is a quadratic irrational. McMullen proved that the Siegel disk for $f$ is self-similar about the critical point. We give a lower bound for the ratio of self-similarity, and we show that if $\theta=(\sqrt 5-1)/2$ is the golden mean, then there exists a triangle contained in the Siegel disk, and with one vertex at the critical point. This answers a 15 year old conjecture.