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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.
In this paper we prove that in any non-trivial real analytic family of unimodal maps, almost any map is either regular (i.e., it has an attracting cycle) or stochastic (i.e., it has an absolutely continuous invariant measure). To this end we show that the space of analytic maps is foliated by codimension-one analytic submanifolds, "hybrid classes". This allows us to transfer the regular or stochastic property of the quadratic family to any non-trivial real analytic family.
The conformal dimension of a metric space is the infimum of the Hausdorff dimensions of all quasisymmetrically equivalent metrics on the space. We show that certain classical self-similar fractal subsets of Euclidean space are not minimal for conformal dimension by constructing explicit metrics in the quasisymmetry class of the Euclidean metric with reduced Hausdorff dimension.
Let $f: [0, +\infty) \longrightarrow (0, +\infty)$ be a sufficiently smooth convex function, vanishing at infinity. Consider the planar domain $Q$ delimited by the positive $x$-semiaxis, the positive $y$-semiaxis, and the graph of $f$. Under certain conditions on $f$, we prove that the billiard flow in $Q$ has a hyperbolic structure and, for some examples, that it is also ergodic. This is done using the cross section corresponding to collisions with the dispersing part of the boundary. The relevant invariant measure for this Poincaré section is infinite, whence the need to surpass the existing results, designed for finite-measure dynamical systems.
We study dynamics of rational maps of degree at least 2 with coefficients in the field $\mathbb{C}_p$, where $p > 1$ is a fixed prime number. The main ingredient is to consider the action of rational maps in $p$-adic hyperbolic space, denoted $\mathbb{H}_p$. Hyperbolic space $\mathbb{H}_p$ is provided with a natural distance, for which it is connected and one dimensional (an $\mathbb{R}$-tree). This advantages with respect to $\mathbb{C}_p$ give new insight into dynamics; in this paper we prove the following results about periodic points. In forthcoming papers we give applications to the Fatou/Julia theory over $\mathbb{C}_p$.
First we prove that the existence of two non-repelling periodic points implies the existence of infinitely many of them. This is in contrast with the complex setting where there can be at most finitely many non-repelling periodic points. On the other hand we prove that every rational map has a repelling fixed point, either in the projective line or in hyperbolic space.
We also caracterise those rational maps with finitely many periodic points in hyperbolic space. Such a rational map can have at most one periodic point (which is then fixed) and we characterise those rational maps having no periodic points and those rational maps having precisely one periodic point in hyperbolic space.
We also prove a formula relating different objects in the projective line and in hyperbolic space, which are fixed by a given rational map. Finally we relate hyperbolic space in the form given here, to well known objects: the Bruhat-Tits building of $PSL(2, \mathbb{C}_p)$ and the Berkovich space of $\mathbb{P}(\mathbb{C}_p)$.
Let $G$ be a real algebraic group defined over $\mathbb{Q}$, let $\Gamma$ be an arithmetic subgroup, and let $T$ be a maximal $\mathbb{R}$-split torus. We classify the closed orbits for the action of $T$ on $G/\Gamma,$ and show that they all admit a simple algebraic description. In particular we show that if $G$ is reductive, an orbit $Tx\Gamma$ is closed if and only if $x^{-1}Tx$ is defined over $\mathbb{Q}$, and is (totally) divergent if and only if $x^{-1}Tx$ is defined over $\mathbb{Q}$ and $\mathbb{Q}$-split. Our analysis also yields the following: there is a compact $K \subset G/\Gamma$ which intersects every $T$-orbit. If $\mathbb{Q} {\rm -rank}(G)<\mathbb{R}{\rm -rank}(G)$, there are no divergent orbits for $T$.
Let $F\colon\mathbb{R}^2 \to \mathbb{R}^2$ be a homeomorphism. An open $F$-invariant subset $U$ of $\mathbb{R}^2$ is a pruning region for $F$ if it is possible to deform $F$ continuously to a homeomorphism $F_U$ for which every point of $U$ is wandering, but which has the same dynamics as $F$ outside of $U$. This concept was motivated by the {\em Pruning Front Conjecture} of Cvitanović, Gunaratne, and Procaccia, which claims that every Hénon map can be understood as a pruned horseshoe. This paper is a survey of pruning theory, concentrating on prunings of the horseshoe. We describe conditions on a disk $D$ which ensure that the orbit of its interior is a pruning region; explain how prunings of the horseshoe can be understood in terms of underlying tree maps; discuss the connection between pruning and Thurston's classification theorem for surface homeomorphisms; motivate a conjecture describing the forcing relation on horseshoe braid types; and use this theory to give a precise statement of the pruning front conjecture.
In this work we deal with non-discrete subgroups of ${\rm Diff}^{\omega} ({\mathbb S}^1)$, the group of orientation-preserving analytic diffeomorphisms of the circle. If $\Gamma$ is such a group, we consider its natural diagonal action ${\widetilde{\Gamma}}$ on the $n-$dimensional torus ${\mathbb T}^n$. It is then obtained a complete characterization of these groups $\Gamma$ whose corresponding ${\widetilde{\Gamma}}-$action on ${\mathbb T}^n$ is not piecewise ergodic (cf. Introduction) for all $n \in {\mathbb N}$ (cf. Theorem A). Theorem A can also be interpreted as an extension of Lie's classification of Lie algebras on ${\mathbb S}^1$ to general non-discrete subgroups of ${\mathbb S}^1$.
Given an orientation-preserving circle homeomorphism $h$, let $(E, \mathcal{L})$ denote a Thurston's left or right earthquake representation of $h$ and $\sigma $ the transversal shearing measure induced by $(E, \mathcal{L})$. We first show that the Thurston norm $||\cdot ||_{Th}$ of $\sigma $ is equivalent to the cross-ratio distortion norm $||\cdot ||_{cr}$ of $h$, i.e., there exists a constant $C>0$ such that $$\frac{1}{C}||h||_{cr}\le ||\sigma ||_{Th} \le C||h||_{cr}$$ for any $h$. Secondly we introduce two new norms on the cross-ratio distortion of $h$ and show they are equivalent to the Thurston norms of the measures of the left and right earthquakes of $h$. Together it concludes that the Thurston norms of the measures of the left and right earthquakes of $h$ and the three norms on the cross-ratio distortion of $h$ are all equivalent. Furthermore, we give necessary and sufficient conditions for the measures of the left and right earthquakes to vanish in different orders near the boundary of the hyperbolic plane. Vanishing conditions on either measure imply that the homeomorphism $h$ belongs to certain classes of circle diffeomorphisms classified by Sullivan in Sullivan.