We review the motivation and fundamental properties of the Hausdorff dimension of metric spaces and illustrate this with a number of examples, some of which are expected and well-known. We also give examples where the Hausdorff dimension has some surprising properties: we construct a set $E\subset\C$ of positive planar measure and with dimension 2 such that each point in $E$ can be joined to $\infty$ by one or several curves in $\C$ such that all curves are disjoint from each other and from $E$, and so that their union has Hausdorff dimension 1. We can even arrange things so that every point in $\C$ which is not on one of these curves is in $E$. These examples have been discovered very recently; they arise quite naturally in the context of complex dynamics, more precisely in the iteration theory of simple maps such as $z\mapsto π\sin(z)$.
In the present paper the classical ideas of Hausdorff and Lebesgue are combined and the Hausdorff--Lebesgue measure is introduced. This makes it possible to obtain new results in chaotic dynamics.
The Hausdorff dimension of a conformal repeller or conformal hyperbolic set is well understood. For non-conformal maps, the Hausdorff dimension is only known in some special cases. Ban, Cao and Hu defined the concept of an average conformal repeller which generalises conformal, quasi-conformal and weakly conformal repellers, and they found an equation for the Hausdorff dimension for an average conformal repeller. In this paper we generalise this concept to average conformal hyperbolic sets, and obtain a similar equation for the Hausdorff dimension.
We construct a self-affine sponge in $\mathbb R^3$ whose dynamical dimension, i.e. the supremum of the Hausdorff dimensions of its invariant measures, is strictly less than its Hausdorff dimension. This resolves a long-standing open problem in the dimension theory of dynamical systems, namely whether every expanding repeller has an ergodic invariant measure of full Hausdorff dimension. More generally we compute the Hausdorff and dynamical dimensions of a large class of self-affine sponges, a problem that previous techniques could only solve in two dimensions. The Hausdorff and dynamical dimens…
It is shown that the boundary of the Mandelbrot set $M$ has Hausdorff dimension two and that for a generic $c \in \bM$, the Julia set of $z \mapsto z^2+c$ also has Hausdorff dimension two. The proof is based on the study of the bifurcation of parabolic periodic points.
Under appropriate assumptions on the dimension of the ambient manifold and the regularity of the Hamiltonian, we show that the Mather quotient is small in term of Hausdorff dimension. Then, we present applications in dynamics.
I show that there exist minimal interval exchange transformations with an ergodic measure whose Hausdorff dimension is arbitrarily small, even 0. I will also show that in particular cases one can bound the Hausdorff dimension between $\frac 1 {2r+4}$ and $\frac 1 r$ for any r greater than 1.
In this paper, we consider two dynamical systems associated to the nearest integer continued fraction, and show that both of them have full Hausdorff dimension spectrum.
We show that at the vicinity of a generic dissipative homoclinic unfolding of a surface diffeomorphism, the Hausdorff dimension of the set of parameters for which the diffeomorphism admits infinitely many periodic sinks is at least 1/2.
We prove that the Hausdorff dimension of an average conformal repeller is stable under random perturbations. Our perturbation model uses the notion of a bundle random dynamical system.
We provide a generalization of the McMullen's algorithm to approximate the Hausdorff dimension of the limit set for convex-cocompact subgroups of isometries of the Complex Hyperbolic Plane.
Given a $k$-self similar set $X\subset [0,1]^{d}$ we calculate both its Hausdorff dimension and its entropy, and show that these two quantities are in fact equal. This affirmatively resolves a conjecture of Adamczewski and Bell.
We study some forward invariant sets appearing in the dynamics of the exponential family. We prove that the Hausdorff dimension of the sets under consideration is not larger than $1$. This allows us to prove, as a consequence, a result for some dynamically defined indecomposable continua which appear in the dynamics of the exponential family. We prove that the Hausdorff dimension of these continua is equal to one.
We show that the supremum for $c$ real of the Hausdorff dimension of the Julia set of the polynomial $z\mapsto z^d+c$ ($d$ is an even natural number) is greater than $2d/(d+1)$.
Let $g$ be a polynomial automorphism of $\C^2$. We study the Hausdorff dimension and topological dimension of the Julia set of $g$. We show that when $g$ is a hyperbolic mapping, then the Hausdorff dimension of the Julia set is strictly greater than its topological dimension. Moreover, the Julia set cannot be locally connected. We also provide estimates for the dimension of the Julia sets in the general (not necessarily hyperbolic) case.
How many fractals exist in nature or the virtual world In this work, we partially answer the second question using Mandelbrots fundamental definition of fractals and their quantities of the Hausdorff dimension and Lebesgue measure. We prove the existence of beth-two of virtual fractals with a Hausdorff dimension of a bivariate function of them and the given Lebesgue measure. The question remains unanswered for other fractal dimensions.
We study non-invertible piecewise hyperbolic maps in the plane. The Hausdorff dimension of the attractor is calculated in terms of the Lyapunov exponents, provided that the map satisfies a transversality condition. Explicit examples of maps for which this condition holds are given.
In this paper, we give a formula of the Hausdorff dimension of the boundary of the immediate basin of infinity of McMullen maps $f_p(z)=z^Q+p/z^Q$, where $Q\geq 3$ and $p$ is small. This gives a lower bound of the Hausdorff dimension of the Julia sets of McMullen maps in the special cases.
The Hausdorff dimension of general Sierpinski carpets, [4] and [20], and the generalization on Lalley-Gatzouras carpets, [10], are today well known results, the formulas being obtain via the variational principle for the dimension. We call the multidimensional versions of these carpets Sierpinski sponges and self-affine sponges, respectively,. In this paper we show that the Hausdorff dimension of self-affine sponges, defined in R3, is a Lipschitz continuous function at Sierpinski sponges.
We show some results about the Hausdorff dimension of particular minimal but not uniquely ergodic interval exchange transformations. There is an appendix which shows that typical points for two different ergodic measures of an interval exchange can approximate each other differently.
In this paper we consider Hausdorff dimension of the sets of Li-Yorke pairs for some chaotic dynamical systems including $A$-coupled expanding systems. We prove that Li-Yorke pairs of $A$- coupled-expanding system under some conditions have full hausdorff dimension on the invariant set. we generalize the result of [8] on the Hausdorff dimension of Li-Yorke pairs of dynamical systems topologically conjugate to the full shift and having a self-similar invariant set to the case of dynamical system conjugated to some kind of subshifts. And Hausdorff dimension of "chaotic invariant set" for some ki…
For one parameter subgroup action on a finite volume homogeneous space, we consider the set of points admitting divergent on average trajectories. We show that the Hausdorff dimension of this set is strictly less than the manifold dimension of the homogeneous space. As a corollary we know that the Hausdorff dimension of the set of points admitting divergent trajectories is not full, which proves a conjecture of Y. Cheung.
Let $d(c)$ denote the Hausdorff dimension of the Julia set $J_c$ of the polynomial $f_c(z)=z^2+c$. We will investigate behavior of the function $d(c)$ when real parameter $c$ tends to a parabolic parameter.
Let $w=(w_1, w_2)$ be a pair of positive real numbers with $w_1+w_2=1$ and $w_1\ge w_2$. We show that the set of $w$-weighted singular vectors in $\mathbb R^2$ has Hausdorff dimension $2- \frac{1}{1+w_1}$. This extends the previous work of Yitwah Cheung on the Hausdorff dimension of the usual (unweighted) singular vectors in $\mathbb R^2$.
We introduce new variants of the notion of geometric pressure for rational functions on the Riemann sphere, including non-hyperbolic functions, in the hope some of them occur useful to achieve a fast approximation from below of the hyperbolic Hausdorff dimension of Julia set.
Using methods from ergodic theory along with properties of the Furstenberg measure we obtain conditions under which certain classes of plane self-affine sets have Hausdorff or box-counting dimensions equal to their affinity dimension. We exhibit some new specific classes of self-affine sets for which these dimensions are equal.
We produce an upper bound for the Hausdorff dimension of the graph of a Weierstrass-type function. Whilst strictly weaker than existing results, it has the advantage of being directly computable from the theory of hyperbolic iterated function systems.
The main result of this paper is to show that if $\H$ is a normal subgroup of a Kleinian group $G$ such that $G/\H$ contains a coset which is represented by some loxodromic element, then the Hausdorff dimension of the transient limit set of $\H$ coincides with the Hausdorff dimension of the limit set of $G$. This observation extends previous results by Fernández and Melián for Riemann surfaces.
We construct a multiply Xiong chaotic set with full Hausdorff dimension everywhere that is contained in some multiply proximal cell for the full shift over finite symbols and the Gauss system respectively.
In this article, we consider hyperbolic rational maps restricted on thier Julia sets and study about the recurrence rate of typical orbits in arbitrarily small neighbourhoods around them and their relationship to the Hausdorff dimension of such small neighbourhoods.
We systematically investigate examples of non-hyperbolic dynamical systems having irregular sets of full topological entropy and full Hausdorff dimension. The examples include some partially hyperbolic systems and geometric Lorenz flows. We also pose interesting questions for future research.
In a recent paper [Adv. Math. 305:165--196, 2017], Komornik et al.~proved a long-conjectured formula for the Hausdorff dimension of the set $\mathcal{U}_q$ of numbers having a unique expansion in the (non-integer) base $q$, and showed that this Hausdorff dimension is continuous in $q$. Unfortunately, their proof contained a gap which appears difficult to fix. This article gives a completely different proof of these results, using a more direct combinatorial approach.
We determine the exact values of Hausdorff dimensions of escaping sets of meromorphic functions with polynomial Schwarzian derivatives. This will follow from the relation between these functions and the second order differential equations in the complex plane.
In this paper, we prove that the Hausdorff dimension of the Rauzy gasket is less than 2. By this result, we answer a question addressed by Pierre Arnoux. Also, this question is a very particular case of the conjecture stated by S.P. Novikov and A. Ya. Maltsev in 2003.
For integer $m\ge3$, we study the dynamical system $(Λ_m,σ_m)$ where $Λ_m$ is the set $\{w\in\{0,1\}^\mathbb{N}: w$ does not contain $0^m$ or $1^m\}$ and $σ_m$ is the shift map on $\{0,1\}^\mathbb{N}$ restricted to $Λ_m$, study the Bernoulli-type measures on $Λ_m$ and find out the unique equivalent $σ_m$-invariant ergodic probability measure. As an application, we obtain the Hausdorff dimension of the set of univoque sequences, the Hausdorff dimension of the set of sequences in which the lengths of consecutive $0$'s and consecutive $1$'s are bounded, and the Hausdorff dimension of their freque…
We consider open billiards in the plane satisfying the no-eclipse condition. We show that the points in the non-wandering set depend differentiably on deformations to the boundary of the billiard. We use Bowen's equation to estimate the Hausdorff dimension of the non-wandering set of the billiard. Finally we show that the Hausdorff dimension depends differentiably on sufficiently smooth deformations to the boundary of the billiard, and estimate the derivative with respect to such deformations.
We compute the Hausdorff dimension of limit sets generated by 3-dimensional self-affine mappings with diagonal matrices of the form A_{ijk}=Diag(a_{ijk}, b_{ij}, c_{i}), where 0<a_{ijk}\le b_{ij}\le c_i<1. By doing so we show that the variational principle for the dimension holds for this class.
We investigate the set of biaccessible points for connected polynomial Julia sets of arbitrary degrees $d\geq 2$. We prove that the Hausdorff dimension of the set of external angles corresponding to biaccessible points is less than 1, unless the Julia set is an interval. This strengthens theorems of Stanislav Smirnov and Anna Zdunik: they proved that the same set of external angles has zero 1-dimensional measure.
A theoretical approach to computing the Hausdorff dimension of the topological boundary of attractors of iterated function systems is developed. The curve known as the Lévy Dragon is then studied in detail and the Hausdorff dimension of its boundary is computed using the theory developed. The actual computation is a complicated procedure. It involves a great deal of combinatorial topology as well as determining the structure and certain eigenvalues of a $752 \times 752$ matrix. Perron-Frobenius theory plays an important role in analyzing this matrix.
In this paper, we show that geometric Lorenz attractors have Hausdorff dimension strictly greater than $2$. We use this result to show that for a "large" set of real functions the Lagrange and Markov Dynamical spectrum associated to these attractors has persistently non-empty interior.
Mean Hausdorff dimension is a dynamical version of Hausdorff dimension. It provides a way to dynamicalize geometric measure theory. We pick up the following three classical results of fractal geometry. (1) The calculation of Hausdorff dimension of homogeneous sets in the circle. (2) The coincidence of Hausdorff and Minkowski dimensions for self-similar sets. (3) The calculation of Hausdorff dimension of Bedford--McMullen carpets. We develop their analogues for mean Hausdorff dimension: (1) The calculation of mean Hausdorff dimension of homogeneous systems in the infinite dimensional torus. (2)…
We consider digits-deleted sets or Cantor-type sets with $β$-expansions. We calculate the Hausdorff dimension $d$ of these sets and show that $d$ is continuous with respect to $β$. The $d$-dimentional Hausdorff measure of these sets is finite and positive.
An approach is given for estimating the Hausdorff dimension of the univoque set of a self-similar set. This sometimes allows us to get the exact Hausdorff dimensions of the univoque sets.
We prove that the Hausdorff dimension of the set of three-period orbits in classical billiards is at most one. Moreover, if the set of three-period orbits has Hausdorff dimension one, then it has a tangent line at almost every point.
We present a construction of hedgehogs for holomorphic maps with an indifferent fixed point. We construct, for a family of commuting non-linearisable maps, a common hedgehog of Hausdorff dimension 1, the minimum possible.
We give an estimate for the Hausdorff dimension of the bifurcation locus of a family of endomorphisms of $\mathbb{P}^k (\mathbb{C})$. This dimension is maximal near isolated Lattès examples.
We show that the Hausdorff dimension of the spectral measure of a class of deterministic, i. e. nonrandom, block-Jacobi matrices may be determined exactly, improving a result of Zlatos (J. Funct. Anal. 207, 216-252 (2004)).
In this note, we show that on certain Gatzouras-Lalley carpet, there exist more than one ergodic measures with full Hausdorff dimension. This gives a negative answer to a conjecture of Gatzouras and Peres.
