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* Starred papers have appeared in the journal cited.
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.
A key point in Douady and Hubbard's study of the Mandelbrot set $M$ is the theorem that every parabolic point $c\ne 1/4$ in $M$ is the landing point for exactly two external rays with angles which are periodic under doubling. This note will try to provide a proof of this result and some of its consequences which relies as much as possible on elementary combinatorics, rather than on more difficult analysis. It was inspired by section 2 of the recent thesis of Schleicher (see also IMS preprint 1994/19, with E. Lau), which contains very substantial simplifications of the Douady-Hubbard proofs with a much more compact argument, and is highly recommended. The proofs given here are rather different from those of Schleicher, and are based on a combinatorial study of the angles of external rays for the Julia set which land on periodic orbits. The results in this paper are mostly well known; there is a particularly strong overlap with the work of Douady and Hubbard. The only claim to originality is in emphasis, and the organization of the proofs.
Consider a compact manifold M of dimension at least 2 and the space of $C^r$-smooth diffeomorphisms $\mathrm{Diff}^r(M)$. The classical Artin-Mazur theorem says that for a dense subset D of $\mathrm{Diff}^r(M)$ the number of isolated periodic points grows at most exponentially fast (call it the A-M property). We extend this result and prove that diffeomorphisms having only hyperbolic periodic points with the A-M property are dense in $\mathrm{Diff}^r(M)$. Our proof of this result is much simpler than the original proof of Artin-Mazur.
The second main result is that the A-M property is not (Baire) generic. Moreover, in a Newhouse domain $\mathcal{N} \subset \mathrm{Diff}^r(M)$, an arbitrary quick growth of the number of periodic points holds on a residual set. This result follows from a theorem of Gonchenko-Shilnikov-Turaev, a detailed proof of which is also presented.
We prove a refinement of the Fatou-Shishikura Inequality - that the total count of nonrepelling cycles of a rational map is less than or equal to the number of independent infinite forward critical orbits - from a suitable application of Thurston's Rigidity Theorem - the injectivity of $I-f_*$ on spaces of meromorphic quadratic differentials.
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.
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.