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 frequency subsets.
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.
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.
Given a positive integer $M$ and a real number $q >1$, a \emph{$q$-expansion} of a real number $x$ is a sequence $(c_i)=c_1c_2\cdots$ with $(c_i) \in \{0,\ldots,M\}^\infty$ such that \[x=\sum_{i=1}^{\infty} c_iq^{-i}.\] It is well known that if $q \in (1,M+1]$, then each $x \in I_q:=\left[0,M/(q-1)\right]$ has a $q$-expansion. Let $\mathcal{U}=\mathcal{U}(M)$ be the set of \emph{univoque bases} $q>1$ for which $1$ has a unique $q$-expansion. The main object of this paper is to provide new characterizations of $\mathcal{U}$ and to show that the Hausdorff dimension of the set of numbers $x \in I…
We fix a positive integer $M$, and we consider expansions in arbitrary real bases $q>1$ over the alphabet $\{0,1,...,M\}$. We denote by $U_q$ the set of real numbers having a unique expansion. Completing many former investigations, we give a formula for the Hausdorff dimension $D(q)$ of $U_q$ for each $q\in (1,\infty)$. Furthermore, we prove that the dimension function $D:(1,\infty)\to[0,1]$ is continuous, and has a bounded variation. Moreover, it has a Devil's staircase behavior in $(q',\infty)$, where $q'$ denotes the Komornik--Loreti constant: although $D(q)>D(q')$ for all $q>q'$, we have $…
Let $K\subseteq\mathbb{R}$ be the unique attractor of an iterated function system. We consider the case where $K$ is an interval and study those elements of $K$ with a unique coding. We prove under mild conditions that the set of points with a unique coding can be identified with a subshift of finite type. As a consequence of this, we can show that the set of points with a unique coding is a graph-directed self-similar set in the sense of Mauldin and Williams \cite{MW}. The theory of Mauldin and Williams then provides a method by which we can explicitly calculate the Hausdorff dimension of thi…
Given a positive integer $M$ and a real number $q>1$, we consider the univoque set $\mathcal{U}_q$ of reals which have a unique $q$-expansion over the alphabet $\set{0,1,\cdots,M}$. In this paper we show that for any $x\in\mathcal{U}_q$ and all sufficiently small $\varepsilon>0$ the Hausdorff dimension $\dim_H\mathcal{U}_q\cap(x-\varepsilon, x+\varepsilon)$ equals either $\dim_H\mathcal{U}_q$ {or} zero. Moreover, we give a complete description of the typical points $x\in\mathcal{U}_q$ which satisfy \[ \dim_H\mathcal{U}_q\cap(x-\varepsilon, x+\varepsilon)=\dim_H\mathcal{U}_q\quad\textrm{for any…
Given $x\in(0, 1]$, let $\mathcal U(x)$ be the set of bases $q\in(1,2]$ for which there exists a unique sequence $(d_i)$ of zeros and ones such that $x=\sum_{i=1}^\infty d_i/q^i$. Lü, Tan and Wu (2014) proved that $\mathcal U(x)$ is a Lebesgue null set of full Hausdorff dimension. In this paper, we show that the algebraic sum $\mathcal U(x)+λ\mathcal U(x)$ and product $\mathcal U(x)\cdot\mathcal U(x)^λ$ contain an interval for all $x\in(0, 1]$ and $λ\ne 0$. As an application we show that the same phenomenon occurs for the set of non-matching parameters studied by the first author and Kalle (20…
Let $\mathbf{J} \subset \mathbb{R}^2$ be the set of couples $(x,q)$ with $q>1$ such that $x$ has at least one representation of the form $x=\sum_{i=1}^{\infty} c_i q^{-i}$ with integer coefficients $c_i$ satisfying $0 \leq c_i < q$, $i \ge 1$. In this case we say that $(c_i)=c_1c_2...$ is an expansion of $x$ in base $q$. Let $\mathbf{U}$ be the set of couples $(x,q) \in \mathbf{J}$ such that $x$ has exactly one expansion in base $q$. In this paper we deduce some topological and combinatorial properties of the set $\mathbf{U}$. We characterize the closure of $\mathbf{U}$, and we determine its H…
Given a positive integer $M$ and a real number $x>0$, let $\mathcal U(x)$ be the set of all bases $q\in(1, M+1]$ for which there exists a unique sequence $(d_i)=d_1d_2\ldots$ with each digit $d_i\in\{0,1,\ldots, M\}$ satisfying $$ x=\sum_{i=1}^\infty\frac{d_i}{q^i}. $$ The sequence $(d_i)$ is called a $q$-expansion of $x$. In this paper we investigate the local dimension of $\mathcal U(x)$ and prove a `variation principle' for unique non-integer base expansions. We also determine the critical values of $\mathcal U(x)$ such that when $x$ passes the first critical value the set $\mathcal U(x)$ c…
Let $K\subset \mathbb{R}$ be a self-similar set generated by some iterated function system. In this paper we prove, under some assumptions, that $K$ can be identified with a subshift of finite type. With this identification, we can calculate the Hausdorff dimension of $K$ as well as the set of elements in $K$ with unique codings using the machinery of Mauldin and Williams \cite{MW}. We give three different applications of our main result. Firstly, we calculate the Hausdorff dimension of the set of points of $K$ with multiple codings. Secondly, in the setting of $β$-expansions, when the set of …
We consider the Hausdorff measure of Julia sets and escaping sets of exponential maps with respect to certain gauge functions. We give conditions on the growth of the gauge function which imply that the measure is zero or infinity, respectively.
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 consider transcendental meromorphic function for which the set of finite singularities of its inverse is bounded. Bergweiler and Kotus gave bounds for the Hausdorff dimension of escaping sets if the function has no logarithmic singularities over infinity, the multiplicities of poles are bounded and the order is finite. We study the case of infinite order and find gauge functions for which the Hausdorff measure of escaping sets is zero or infinity.
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.
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.
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 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 $2d/(d+1)$.
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.
Given positive integers $\ell<n$ and a real $d\in (\ell,n)$, we construct sets $K\subset \mathbb R^n$ with positive and finite Hausdorff $d-$measure such that the Radon-Nikodym derivative associated to all projections on $\ell-$dimensional planes is not an $L^p$ function, for all $p>1$.
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.
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.
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.
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.
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 closed interval and circle are the only smooth Julia sets in polynomial dynamics. D. Ruelle proved that the Hausdorff dimension of unicritical Julia sets close to the circle depends analytically on the parameter. Near the tip of the Mandelbrot set M, the Hausdorff dimension is generally discontinuous. Answering a question of J-C. Yoccoz in the conformal setting, we observe that the Hausdorff dimension of quadratic Julia sets depends continuously on $c$ and find explicit bounds at the tip of M for most real parameters in the the sense of 1-dimensional Lebesgue measure.
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…
In this paper we study the dimension of a family of sets arising in open dynamics. We use exponential mixing results for diagonalizable flows in compact homogeneous spaces $X$ to show that the Hausdorff dimension of set of points that lie on trajectories missing a particular open ball of radius $r$ is at most $$\dim X + C\frac{r^{\dim X}}{\log r},$$ where $C>0$ is a constant independent of $r>0$. Meanwhile, we also describe a general method for computing the least cardinality of open covers of dynamical sets using volume estimates.
We show that the escaping sets and the Julia sets of bounded type transcendental entire functions of order $ρ$ become 'smaller' as $ρ\to\infty$. More precisely, their Hausdorff measures are infinite with respect to the gauge function $h_γ(t)=t^2g(1/t)^γ$, where $g$ is the inverse of a linearizer of some exponential map and $γ\geq(\logρ(f)+K_1)/c$, but for $ρ$ large enough, there exists a function $f_ρ$ of bounded type with order $ρ$ such that the Hausdorff measures of the escaping set and the Julia set of $f_ρ$ with respect to $h_{γ'}$ are zero whenever $γ'\leq(\logρ-K_2)/c$.
We consider linear mappings on the $d$-dimensional torus, defined by $T(x) = Ax \pmod 1$, where $A$ is an invertible $d \times d$ integer matrix, with no eigenvalues on the unit circle. In the case $d = 2$ and $\det A = \pm 1$, we give a formula for the Hausdorff dimension of the set \[ \{ \, x \in \mathbb{T}^d : d (T^n (x), x) < e^{- αn} \text{ for infinitely many } n \, \}. \]
We show that if $β>1$ is a rational number and the Julia set $J$ of the holomorphic correspondence $z^β+c$ is a locally eventually onto hyperbolic repeller, then the Hausdorff dimension of $J$ is bounded from above by the zero of the associated pressure function. As a consequence, we conclude that the Julia set of the correspondence has zero Lebesgue measure for parameters close to zero, whenever $q^2<p$ and $β=p/q$ in lowest terms.
We prove that for every polynomial of one complex variable of degree at least 2 and Julia set not being totally disconnected nor a circle, nor interval, Hausdorff dimension of this Julia set is larger than 1. Till now this was known only in the connected Julia set case. We give also an example of a polynomial with non-connected but not totally disconnected Julia set and such that all its components comprising of more than single points are analytic arcs, thus resolving a question by Christopher Bishop, who asked whether every such component must have Hausdorff dimension larger than 1.
In this paper we prove that among pairs $K,\,K' \subset \mathbb{C}$ of conformal dynamically defined Cantor sets with sum of Hausdorff dimensions $HD(K)+HD(K')>2$, there is an open and dense subset of such pairs verifying $\text{int}(K-K')\neq \emptyset$. This is motivated by the work \cite{MY}, where Moreira and Yoccoz proved a similar statement for dynamically defined Cantor sets in the real line. Here we adapt their argument to the context of conformal Cantor sets in the complex plane, this requires the introduction of several new concepts and a more detailed analysis in some parts of the a…
In 1994, Kuperberg constructed a smooth flow on a three-manifold with no periodic orbits. It was later shown that a generic Kuperberg flow preserves a codimension one laminar minimal set. We develop new techniques to study the symbolic dynamics and dimension theory of this minimal set, by relating it to the limit set of a graph directed pseudo-Markov system over a countable alphabet.
We exhibit a class of Schottky subgroups of $\mathbf{PU}(1,n)$ ($n \geq 2$) which we call well-positioned and show that the Hausdorff dimension of the limit set $Λ_Γ$ associated with such a subgroup $Γ$, with respect to the spherical metric on the boundary of complex hyperbolic $n$-space, is equal to the growth exponent $δ_Γ$. For general $Γ$ we establish (under rather mild hypotheses) a lower bound involving the dimension of the Patterson-Sullivan measure along boundaries of complex geodesics. Our main tool is a version of the celebrated Ledrappier-Young theorem.
Given a finite set $\mathcal{A} \subseteq \mathrm{SL}(2,\mathbb{R})$ we study the dimension of the attractor $K_\mathcal{A}$ of the iterated function system induced by the projective action of $\mathcal{A}$. In particular, we generalise a recent result of Solomyak and Takahashi by showing that the Hausdorff dimension of $K_\mathcal{A}$ is given by the minimum of 1 and the critical exponent, under the assumption that $\mathcal{A}$ satisfies certain discreteness conditions and a Diophantine property. Our approach combines techniques from the theories of iterated function systems and Möbius semig…
Certain subsets of limit sets of geometrically finite Fuchsian groups with parabolic elements are considered. It is known that Jarn\'ık limit sets determine a "weak multifractal spectrum" of the Patterson measure in this situation. This paper will describe a natural generalisation of these sets, called strict Jarn\'ık limit sets, and show how these give rise to another weak multifractal spectrum. Number-theoretical interpretations of these results in terms of continued fractions will also be given.
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 study the Hausdorff dimension and measures of full Hausdorff dimension for a compact invariant set of an expanding nonconformal map on the torus given by an integer-valued diagonal matrix. The Hausdorff dimension of a "general Sierpinski carpet" was found by McMullen and Bedford and the uniqueness of the measure of full Hausdorff dimension in some cases was proved by Kenyon and Peres. We extend these results by using compensation functions to study a general Sierpinski carpet represented by a shift of finite type. We give some conditions under which a general Sierpinski carpet has a unique …
This article concerns a class of open billiards consisting of a finite number of strictly convex, non-eclipsing obstacles $K$. The non-wandering set $M_0$ of the billiard ball map is a topological Cantor set and its Hausdorff dimension has been previously estimated for billiards in $\mathbb{R}^2$, using well-known techniques. We extend these estimates to billiards in $\mathbb{R}^n$, and make various refinements to the estimates. These refinements also allow improvements to other results. We also show that in many cases, the non-wandering set is confined to a particular subset of $\mathbb{R}^n$…
Let $(Σ, σ)$ be the one-sided shift space with $m$ symbols and $R_n(x)$ be the first return time of $x\inΣ$ to the $n$-th cylinder containing $x$. Denote $$E^φ_{α,β}=\left\{x\inΣ: \liminf_{n\to\infty}\frac{\log R_n(x)}{φ(n)}=α,\ \limsup_{n\to\infty}\frac{\log R_n(x)}{φ(n)}=β\right\},$$ where $φ: \mathbb{N}\to \mathbb{R}^+$ is a monotonically increasing function and $0\leqα\leqβ\leq +\infty$. We show that the Hausdorff dimension of the set $E^φ_{α,β}$ admits a dichotomy: it is either zero or one depending on $φ, α$ and $β$.
We show that contrary to anticipation suggested by the dictionary between rational maps and Kleinian groups and by the ``hairiness phenomenon'', there exist many Feigenbaum Julia sets $J(f)$ whose Hausdorff dimension is strictly smaller than two. We also prove that for any Feigenbaum Julia set, the Poincaré critical exponent $\de_\crit$ is equal to the hyperbolic dimension $\HD_\hyp(J(f))$. Moreover, if $\area J(f)=0$ then $\HD_\hyp (J(f))=\HD(J(f))$. In the stationary case, the last statement can be reversed: if $\area J(f)> 0$ then $\HD_\hyp (J(f))< 2$. We also give a new construction of con…
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.
In this article, we study the Hausdorff measure of shrinking target sets on self-conformal sets. The Hausdorff dimension of the sets we are interested in here was established by Hill and Velani in 1995. However, until recently, little more was known about the Hausdorff measure of these particular sets. In this paper we provide a complete characterisation of the Hausdorff measure of these sets, obtaining a dichotomy for the Hausdorff measure which is determined by the convergence or divergence of a sum depending on the radii of our "shrinking targets". Our main result complements earlier work o…
We consider (not self-similar) Cantor sets defined by a sequence of piecewise linear functions. We prove that the dimension of the harmonic measure on such a set is strictly smaller than its Hausdorff dimension. Some Hausdorff measure estimates for these sets are also provided.
