Hausdorff Dimension Of Limit Sets Research Articles

On the Dimension of Limit Sets on ℙ(ℝ3) via Stationary Measures: Variational Principles and Applications

Abstract This paper investigates the (semi)group action of $\textrm{SL}_{n}({\mathbb R})$ on ${\mathbb P}({\mathbb R}^{n})$, a primary example of non-conformal, non-linear, and non-strictly contracting action. We establish variational principles of the affinity exponent for two main examples: the Borel Anosov representations and the Rauzy gasket. In [ 32], they obtain a dimension formula for the stationary measures on ${\mathbb P}({\mathbb R}^{3})$. Combined with our result, it allows us to study the Hausdorff dimension of limit sets of Anosov representations in $\textrm{SL}_{3}({\mathbb R})$ and the Rauzy gasket. It yields the equality between the Hausdorff dimensions and the affinity exponents in both settings, generalizing the classical Patterson–Sullivan formula. In the appendix, we improve the numerical lower bound of the Hausdorff dimension of Rauzy gasket to $1.5$.

Hausdorff dimension of limit sets for projective Anosov representations

Hausdorff dimension of limit sets for projective Anosov representations

Conformality for a robust class of non-conformal attractors

Abstract In this paper we investigate the Hausdorff dimension of limit sets of Anosov representations. In this context we revisit and extend the framework of hyperconvex representations and establish a convergence property for them, analogue to a differentiability property. As an application of this convergence, we prove that the Hausdorff dimension of the limit set of a hyperconvex representation is equal to a suitably chosen critical exponent.

Basmajian-type identities and Hausdorff dimension of limit sets

In this paper, we study Basmajian-type series identities on holomorphic families of Cantor sets associated to one-dimensional complex dynamical systems. We show that the series is absolutely summable if and only if the Hausdorff dimension of the Cantor set is strictly less than one. Throughout the domain of convergence, these identities can be analytically continued and they exhibit non-trivial monodromy.

Finer fractal geometry for analytic families of conformal dynamical systems

We prove several results establishing real analyticity of Hausdorff dimensions of limit sets of analytic families of conformal graph directed Markov systems. With this tool and with iterated function systems resulting from the existence of nice sets in the sense of Rivera-Letelier, we prove that the canonical Hausdorff measure restricted to the radial Julia set of a tame meromorphic function (can be rational) is σ-finite and that the Hausdorff dimension of the radial Julia sets for fairly general families of meromorphic functions (can be rational) is real analytic.

Graph directed Markov systems on Hilbert spaces

AbstractWe deal with contracting finite and countably infinite iterated function systems acting on Polish spaces, and we introduce conformal Graph Directed Markov Systems on Polish spaces. Sufficient conditions are provided for the closure of limit sets to be compact, connected, or locally connected. Conformal measures, topological pressure, and Bowen's formula (determining the Hausdorff dimension of limit sets in dynamical terms) are introduced and established. We show that, unlike the Euclidean case, the Hausdorff measure of the limit set of a finite iterated function system may vanish. Investigating this issue in greater detail, we introduce the concept of geometrically perfect measures and provide sufficient conditions for geometric perfectness. Geometrical perfectness guarantees the Hausdorff measure of the limit set to be positive. As a by–product of the mainstream of our investigations we prove a 4r–covering theorem for all metric spaces. It enables us to establish appropriate co–Frostman type theorems.

Analytic families of semihyperbolic generalized polynomial-like mappings

We show that the Hausdorff dimension of Julia sets in any analytic family of semihyperbolic generalized polynomial-like mappings (GPL) depends in a real-analytic manner on the parameter. For the proof we introduce abstract weakly regular analytic families of conformal graph directed Markov systems. We show that the Hausdorff dimension of limit sets in such families is real-analytic, and we associate to each analytic family of semihyperbolic GPLs a weakly regular analytic family of conformal graph directed Markov systems with the Hausdorff dimension of the limit sets equal to the Hausdorff dimension of the Julia sets of the corresponding semihyperbolic GPLs.

Dimension sets for infinite IFSs: the Texan Conjecture

We consider the set of Hausdorff dimensions of limit sets of finite subsystems of an infinite conformal iterated function system and refer to it as the restricted dimension set. The corresponding set for all subsystems will be referred to as the complete dimension set. We give sufficient conditions for a point to belong to the complete dimension set and consequently to be an accumulation point of the restricted dimension set. We also give sufficient conditions on the system for both sets to be nowhere dense in some interval. Both general results are illustrated by examples. Applying the first result to the case of continued fraction we are able to prove the Texan Conjecture, that is we show that the set of Hausdorff dimensions of bounded type continued fraction sets is dense in the unit interval.

Hausdorff dimension of limit sets of discrete subgroups of higher rank Lie groups

Let X be a globally symmetric space of noncompact type, \( G = \textrm{Isom}^o (X) \) and \( \Gamma \subset G \) a discrete subgroup. Introducing an appropriate notion of Hausdorff measure on the geometric boundary \( \theta X \) of \( \theta X \), we prove that for regular boundary points \( \xi \in \theta X \) , the Hausdorff dimension of the radial limit set in \( G \cdot \xi \) is bounded above by the exponential growth rate of the number of orbit points close in direction to \( G \cdot \xi \subseteq \theta X \). Furthermore, for Zariski dense discrete groups Γ we construct Γ-invariant densities with support in every G-invariant subset of the limit set and study their properties. For a class of groups which generalises convex cocompact groups in the rank one setting, these densities allow to give a sharp estimate on the Hausdorff dimension of the radial limit set in each subset \( G \cdot \xi \subseteq \theta X \).

Hausdorff dimension of limit sets for parabolic IFS with overlaps

We study parabolic iterated function systems with overlaps on the real line. We show that if a d-parameter family of such systems satisfies a transversality condition, then for almost every parameter value the Hausdorff dimension of the limit set is the minimum of 1 and the least zero of the pressure function. Moreover, the local dimension of the exceptional set of parameters is estimated. If the least zero is greater than 1, then the limit set (typically) has positive Lebesgue measure. These results are applied to some specific families including those arising from a class of continued fractions.

Length distortion and the Hausdorff dimension of limit sets

Let Γ be a convex co-compact quasi-Fuchsian Kleinian group. We define the distortion function along geodesic rays lying on the boundary of the convex hull of the limit set, where each ray is pointing in a randomly chosen direction. The distortion function measures the ratio of the intrinsic to extrinsic metrics, and is defined asymptotically as the length of the ray goes to infinity. Our main result is that the distortion function is both almost everywhere constant and bounded above by the Hausdorff dimension of the limit set of Γ. As a consequence, we are able to provide a geometric proof of the following result of Bowen: If the limit set of Γ is not a round circle, then the Hausdorff dimension of the limit set is strictly greater than one. The proofs are developed from results in Patterson-Sullivan theory and ergodic theory.

Spectral theory, Hausdorff dimension and the topology of hyperbolic 3-manifolds

Let M be a compact 3-manifold whose interior admits a complete hyperbolic structure. We let Λ(M) be the supremum of λ0(N) where N varies over all hyperbolic 3-manifolds homeomorphic to the interior of N. Similarly, we let D(M) be the infimum of the Hausdorff dimensions of limit sets of Kleinian groups whose quotients are homeomorphic to the interior of M. We observe that A(M) = D(M)(2 − D(M)) if M is not handlebody or a thickened torus. We characterize exactly when A(M) = 1 and D(M) = 1 in terms of the characteristic submanifold of the incompressible core of M.

Hausdorff dimension and conformal dynamics, III: Computation of dimension

This paper presents an eigenvalue algorithm for accurately computing the Hausdorff dimension of limit sets of Kleinian groups and Julia sets of rational maps. The algorithm is applied to Schottky groups, quadratic polynomials and Blaschke products, yielding both numerical and theoretical results. Dimension graphs are presented for (a) the family of Fuchsian groups generated by reflections in 3 symmetric geodesics; (b) the family of polynomials f c ( z ) = z 2 + c, c ∈ [-1, 1/2]; and (c) the family of rational maps f t ( z ) = z / t + 1/ z, t ∈ (0, 1]. We also calculate H.dim (Λ) ≈ 1.305688 for the Apollonian gasket, and H.dim ( J ( f )) ≈ 1.3934 for Douady's rabbit, where f ( z ) = z 2 + c satisfies f 3 (0) = 0.

Hausdorff Dimension of Limit Sets for Spherical CR Manifolds

LetM2n+1(n⩾1) be a compact, spherical CR manifold. SupposeM2n+1is its universal cover andΦ:M2n+1→S2n+1is on injective CR developing map, whereS2n+1is the standard unit sphere in the complex (n+1)-spaceCn+1, thenM2n+1is of the quotient formΩ/Λ, whereΩis a simply connected open set inS2n+1, andΛis a complex Klein group acting onΩproperly discontinuously. In this paper, we show that if the CR Yamabe invariant ofM2n+1is positive, then the Carnot Hausdorff dimension of the limit set ofΛis bounded above byn·s(M2n+1), wheres(M2n+1)⩽1 and is a CR invariant. The method that we adopt is analysis of the CR invariant Laplacian. We also explain the geometric origin of this question.

Hausdorff dimensions of limit sets I

Hausdorff dimensions of limit sets I

On the calculation of quantum mechanical ground states from classical geodesic motion on certain spaces of constant negative curvature

We consider geodesic motion on three-dimensional Riemannian manifolds of constant negative curvature, topologically equivalent to S × ]0, 1[, S a compact surface of genus two. To those trajectories which are bounded and recurrent in both directions of the time evolution t → + ∞, t → − ∞ a fractal limit set is associated whose Hausdorff dimension is intimately connected with the quantum mechanical energy ground state, determined by the Schrödinger operator on the manifold. We give a rather detailed and pictorial description of the hyperbolic spaces we have in mind, discuss various aspects of classical and quantum mechanical motion on them as far as they are needed to establish the connection between energy ground state and Hausdorff dimension and give finally some examples of ground state calculations in terms of Hausdorff dimensions of limit sets of classical trajectories.

The Hausdorff dimension of limit sets of fuchsian groups of the type ( g;m)

1. Preliminaries. Let G be a non-elementary finitely generated Fuchsian group of the second kind acting on the unit disc Δ, and let Λ( G ) be the limit set of G. It is well known that Λ( G ) is a perfect non-dense set on the boundary of A. The Hausdorff dimension of the limit set Λ( G ) of G is defined as the non-negative number where m t (Λ( G )) denotes the t -dimensional measure of Λ( G ) ([2], [3]). We say G has the type (g; m) if S = Δ/ G is obtained from a compact surface of genus g by removing m (≥ 0) conformal discs.

The Hausdorff dimension of limit sets of some Fuchsian groups

The Hausdorff dimension of limit sets of some Fuchsian groups