Yun Zhao, Yongluo Cao, Jungchao Ban
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.
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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.
Bowen's formula relates the Hausdorff dimension of a conformal repeller to the zero of a `pressure' function. We present an elementary, self-contained proof which bypasses measure theory and the Thermodynamic Formalism to show that Bowen's formula holds for $C^1$ conformal repellers. We consider time-dependent conformal repellers obtained as invariant subsets for sequences of conformally expanding maps within a suitable class. We show that Bowen's formula generalizes to such a repeller and that if the sequence is picked at random then the Hausdorff dimension of the repeller almost surely agree…
Given a non-conformal repeller $Λ$ of a $C^{1+γ}$ map, we study the Hausdorff dimension of the repeller and continuity of the sub-additive topological pressure for the sub-additive singular valued potentials. Such a potential always possesses an equilibrium state. We then use a substantially modified version of Katok's approximating argument, to construct a compact invariant set on which the corresponding dynamical quantities (such as Lyapunov exponents and metric entropy) are close to that of the equilibrium measure. This allows us to establish continuity of the sub-additive topological press…
We prove the existence of an ergodic measure with full Hausdorff dimension for a class of nonlinear nonconformal skew-product transformations. In order to do so we establish a variational principle for the topological pressure of certain noncompact sets.
We give a subclass L of Non-linear Lalley-Gatzouras carpets and an open set U in L such that any carpet in U has a unique ergodic measure of full dimension. In particular, any Lalley-Gatzouras carpet which is close to a non-trivial general Sierpinski carpet has a unique ergodic measure of full dimension.
We consider the multifractal analysis for Birkhoff averages of continuous potentials on a class of non-conformal repellers corresponding to the self-affine limit sets studied by Lalley and Gatzouras. A conditional variational principle is given for the Hausdorff dimension of the set of points for which the Birkhoff averages converge to a given value. This extends a result of Barral and Mensi to certain non-conformal maps with a measure dependent Lyapunov exponent.
Dimensions of surface repellers and attractors of non-linear planar IFSs
该研究证明了$C^r$通用表面排斥子的维数公式，推广了满足强分离条件的平面非线性迭代函数系统吸引子的维数结果，并弱化了Weierstrass型函数图像二分结果的正则性要求。
The dimension of the non-dense orbit set for expanding maps
本文基于Cao、Pesin和Zhao给出的排斥子Hausdorff维数精确下界，为膨胀映射的非稠密轨道集的Hausdorff维数提供了新证明，并证明其Carathéodory奇异维数与排斥子相等。
Michael Barnsley introduced a family of fractals sets which are repellers of piecewise affine systems. The study of these fractals was motivated by certain problems that arose in fractal image compression but the results we obtained can be applied for the computation of the Hausdorff dimension of the graph of some functions, like generalized Takagi functions and fractal interpolation functions. In this paper we introduce this class of fractals and present the tools in the one-dimensional dynamics and nonconformal fractal theory that are needed to investigate them. This is the first part in a s…
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.
New bounds for the Hausdorff dimension of a dynamically defined Cantor set
本文利用加性热力学形式，为光滑流形上分段扩张映射定义的非共形双曲排斥子推导出 Hausdorff 维数与计盒维数的新界。
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…
The pressure function $P(A, s)$ plays a fundamental role in the calculation of the dimension of "typical" self-affine sets, where $A=(A_1,\ldots, A_k)$ is the family of linear mappings in the corresponding generating iterated function system. We prove that this function depends continuously on $A$. As a consequence, we show that the dimension of "typical" self-affine sets is a continuous function of the defining maps. This resolves a folklore open problem in the community of fractal geometry. Furthermore we extend the continuity result to more general sub-additive pressure functions generated …
Continuity of Hausdorff Dimension at Hopf Bifurcation
研究通过Hopf分岔从传递Anosov微分同胚得到的非双曲排斥子的Hausdorff维数和盒维数在参数分岔处的连续性，通过带孔映射方法证明。
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.
On orbit complexity of dynamical systems: intermediate value property and level set related to a Furstenberg problem
本文在温和条件下研究符号动力系统的轨道复杂度，解决了子系统拓扑熵降低问题，确定了给定复杂度水平集及Furstenberg水平集的Hausdorff维数，并应用于β-变换和共形扩张排斥子等系统。
We study several fractal properties of the Weierstrass-type function \[ W(x)=\sum_{n=0} ^\infty λ(x) λ(τx) \cdots λ(τ^{n-1}x)\, g(τ^n x), \] where $τ:[0,1)\to[0,1)$ is a cookie cutter map with possibly fractal repeller, and $λ$ and $g$ are functions with proper regularity. In the first part, we determine the box dimension of the graph of $W$ and Hausdorff dimension of its randomised version. In the second part, the Housdorff spectrum of the local Hölder exponent is characterised in terms of thermodynamic formalisms. Furthermore, in the randomised case, a novel formula for the lifted Hausdorff …
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…
The extended Hausdorff dimension spectrum of a conformal iterated function system is maximal
本文证明任意共形迭代函数系统的扩展 Hausdorff 维数谱是极大的，即所有不超过极限集维数的非负维数均可实现，关键步骤是证明 β-移位族极限集的维数连续依赖于 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…
Hausdorff dimension of the set of eventually always hitting points on a self-conformal set
本文在自共形迭代函数系统框架下研究最终总是命中集与收缩目标集之交的豪斯多夫维数，给出其上界和下界，并得到两者交集的维数结果。
Dimensions and dimension spectra of Non-autonomous iterated function systems
本文研究非自主共形集的中间维度谱，通过拓扑压力理论推导其豪斯多夫维数和盒计数维数，并探讨无限共形迭代函数系统的维数公式。
We extend results by B. Hasselblatt, J. Schmeling in \emph{Dimension product structure of hyperbolic sets} (2004), and by the third author and K. Simon in \emph{Hausdorff and packing measures for solenoids} (2003), for $C^{1+\varepsilon}$ hyperbolic, (partially) linear solenoids $Λ$ over the circle embedded in $\mathbb{R}^3$ non-conformally attracting in the stable discs $W^s$ direction, to nonlinear ones. Under an assumption of transversality and assumptions on Lyapunov exponents for an appropriate Gibbs measure imposing \emph{thinness}, assuming also there is an invariant $C^{1+\varepsilon}$…
This paper presents a simple method of calculating the Hausdorff dimension for a class of non-conformal fractals.
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 investigate the Hausdorff measure and content on a class of quasi self-similar sets that include, for example, graph-directed and sub self-similar and self-conformal sets. We show that any Hausdorff measurable subset of such a set has comparable Hausdorff measure and Hausdorff content. In particular, this proves that graph-directed and sub self-conformal sets with positive Hausdorff measure are Ahlfors regular, irrespective of separation conditions. When restricting to self-conformal subsets of the real line with Hausdorff dimension strictly less than one, we additionally show that the weak…
Projections of Gibbs measures on self-conformal sets
本文在旋转斜积遍历性假设下，证明自共形集上吉布斯测度的正交投影具有最大豪斯多夫维数，无需分离条件，并推出Falconer距离集猜想的维数形式。
Shrinking targets versus recurrence: A survey on the Hausdorff dimension theory
该综述对比动力系统中收缩目标集与回归集的豪斯多夫维数，涉及多类动力系统，还涵盖相关测度结果及坏逼近轨道等内容。
This paper considers self-conformal iterated function systems (IFSs) on the real line whose first level cylinders overlap. In the space of self-conformal IFSs, we show that generically (in topological sense) if the attractor of such a system has Hausdorff dimension less than $1$ then it has zero appropriate dimensional Hausdorff measure and its Assouad dimension is equal to $1$. Our main contribution is in showing that if the cylinders intersect then the IFS generically does not satisfy the weak separation property and hence, we may apply a recent result of Angelevska, Käenmäki and Troscheit […
We highlight a connection between Diophantine approximation and the lower Assouad dimension by using information about the latter to show that the Hausdorff dimension of the set of badly approximable points that lie in certain non-conformal fractals, known as self-affine sponges, is bounded below by the dynamical dimension of these fractals. In particular, for self-affine sponges with equal Hausdorff and dynamical dimensions, the set of badly approximable points has full Hausdorff dimension in the sponge. Our results, which are the first to advance beyond the conformal setting, encompass both …
We consider small perturbations of a conformal iterated function system (CIFS) produced by either adding or removing some generators with small derivative from the original. We establish a formula, utilizing transfer operators arising from the thermodynamic formalism à la Sinai--Ruelle--Bowen, which may be solved to express the Hausdorff dimension of the perturbed limit set in series form: either exactly, or as an asymptotic expansion. Significant applications include strengthening Hensley's asymptotic formula from 1992, which improved on earlier bounds due to Jarník and Kurzweil, for the Haus…
We show that the Julia set of quadratic maps with parameters in hyperbolic components of the Mandelbrot set is given by a transseries formula, rapidly convergent at any repelling periodic point. Up to conformal transformations, we obtain $J$ from a smoother curve of lower Hausdorff dimension, by replacing pieces of the more regular curve by increasingly rescaled elementary "bricks" obtained from the transseries expression. Self-similarity of $J$, up to conformal transformation, is manifest in the formulas. The Hausdorff dimension of $J$ is estimated by the transseries formula. The analysis ext…
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.
We consider a generalisation of the self-affine iterated function systems of Lalley and Gatzouras by allowing for a countable infinity of non-conformal contractions. It is shown that the Hausdorff dimension of the limit set is equal to the supremum of the dimensions of compactly supported ergodic measures. In addition we consider the multifractal analysis for countable families of potentials. We obtain a conditional variational principle for the level sets.
We show that self-conformal subsets of $\mathbb{R}$ that do not satisfy the weak separation condition have full Assouad dimension. Combining this with a recent results by Käenmäki and Rossi we conclude that an interesting dichotomy applies to self-conformal and not just self-similar sets: if $F\subset\mathbb{R}$ is self-conformal with Hausdorff dimension strictly less than $1$, either the Hausdorff dimension and Assouad dimension agree or the Assouad dimension is $1$. We conclude that the weak separation property is in this case equivalent to Assouad and Hausdorff dimension coinciding. (This m…
We consider the family of CIFSs of generalized complex continued fractions with a complex parameter space. This is a new interesting example to which we can apply a general theory of infinite CIFSs and analytic families of infinite CIFSs. We show that the Hausdorff dimension function of the family of the CIFSs of generalized complex continued fractions is continuous in the parameter space and is real-analytic and subharmonic in the interior of the parameter space. As a corollary of these results, we also show that the Hausdorff dimension function has a maximum point and the maximum point belon…
Rigorous Hausdorff dimension estimates for conformal fractals
本文开发了一个通用框架，通过有限元离散化和迭代网格方案对 Perron-Frobenius 算子特征函数进行分段线性近似，严格估计了 $\mathbb{R}^n$ 中极大共形图定向 Markov 系统的 Hausdorff 维数，并提供了特征函数导数界限。
Let $f: M \to M$ be a $C^{1+α}$ map/diffeomorphism of a compact Riemannian manifold $M$ and $μ$ be an expanding/hyperbolic ergodic $f$-invariant Borel probability measure on $M$. Assume $f$ is average conformal expanding/hyperbolic on the support set $W$ of $μ$ and $W$ is locally maximal. For any subset $A\subset W$ with small entropy or dimension, we investigate the topological entropy and Hausdorff dimensions of the $A$-exceptional set and the limit $A$-exceptional set.
We consider a class of non-conformal expanding maps on the $d$-dimensional torus. For an equilibrium measure of an Hölder potential, we prove an analogue of the Central Limit Theorem for the fluctuations of the logarithm of the measure of balls as the radius goes to zero. An unexpected consequence is that when the measure is not absolutely continuous, then half of the balls of radius $\eps$ have a measure smaller than $\eps^δ$ and half of them have a measure larger than $\eps^δ$, where $δ$ is the Hausdorff dimension of the measure. We first show that the problem is equivalent to the study of t…
We study rational functions satisfying summability conditions - a family of weak conditions on the expansion along the critical orbits. Assuming their appropriate versions, we derive many nice properties: There exists a unique, ergodic, and non-atomic conformal measure with exponent equal to the Hausdorff dimension of the Julia set. There is an absolutely continuous invariant measure with respect to this conformal measure. The Minkowski dimension of the Julia set is strictly less than 2. Either the Julia set is the whole sphere, or the dynamics is unstable. For such polynomials and Blaschke pr…
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.
This paper introduces the concept of average conformal hyperbolic sets, which admit only one positive and one negative Lyapunov exponents for any ergodic measure. For an average conformal hyperbolic set of a C1 diffeomorphism, utilizing the techniques in sub-additive thermodynamics formalism and some geometric arguments with unstable/stable manifolds, a formula of the Hausdorff dimension and lower (upper) box dimension is given in this paper, which are exactly the sum of the dimensions of the restriction of the hyperbolic set to a stable and unstable manifolds. Furthermore, the dimensions of a…
We study continuity and discontinuity properties of some popular measure-dimension mappings under some topologies on the space of probability measures in this work. We give examples to show that no continuity can be guaranteed under general weak, setwise or TV topology on the space of measures for any of these measure-dimension mappings. However, in some particular circumstances or by assuming some restrictions on the measures, we do have some (semi-)continuity results. We then apply our continuity results to concerning measures appearing in two kinds of infinite iterated function systems, nam…
The boundaries of the hyperbolic components of odd period of the multicorns contain real-analytic arcs consisting of quasi-conformally conjugate parabolic parameters. One of the main results of this paper asserts that the Hausdorff dimension of the Julia sets is a real-analytic function of the parameter along these parabolic arcs. This is achieved by constructing a complex one-dimensional quasiconformal deformation space of the parabolic arcs which are contained in the dynamically defined algebraic curves $\mathrm{Per}_n(1)$ of a suitably complexified family of polynomials. As another applicat…
The dimension spectrum of the infinitely generated Apollonian gasket
本文通过计算机辅助证明，结合迭代技术与严格估计豪斯多夫维数的方法，证明了无限生成的阿波罗垫片具有完整的维数谱。
Given a rational map of the Riemann sphere and a subset $A$ of its Julia set, we study the $A$-exceptional set, that is, the set of points whose orbit does not accumulate at $A$. We prove that if the topological entropy of $A$ is less than the topological entropy of the full system then the $A$-exceptional set has full topological entropy. Furthermore, if the Hausdorff dimension of $A$ is smaller than the dynamical dimension of the system then the Hausdorff dimension of the $A$-exceptional set is larger than or equal to the dynamical dimension, with equality in the particular case when the dyn…
We study non-autonomous conformal iterated function systems, with finite or countably infinite alphabet alike. These differ from the usual (autonomous) iterated function systems in that the contractions applied at each step in time are allowed to vary. (In the case where all maps are affine similarities, the resulting system is also called a "Moran set construction".) We shall show that, given a suitable restriction on the growth of the number of contractions used at each step, the Hausdorff dimension of the limit set of such a system is determined by an equation known as Bowen's formula. We a…
On the Hausdorff dimension and Cantor set structure of sliding Shilnikov invariant sets
本文对Filippov系统中Shilnikov滑动连接的首回归映射进行局部分析，揭示了其共形迭代函数系统结构，证明了其局部不变集的闭包为Cantor集，并精确估计了其Hausdorff维数严格介于0和1之间且Lebesgue测度为0。
The dimension spectrum of a conformal iterated function system (CIFS) is the set of all Hausdorff dimensions of its various subsystem limit sets. This brief note provides two constructions -- (i) a compact perfect set that cannot be realized as the dimension spectrum of a CIFS; and (ii) a similarity IFS whose dimension spectrum has zero Hausdorff dimension, and thus is not uniformly perfect -- which resolve questions posed by Chousionis, Leykekhman, and Urbański, and go on provoke fresh conjectures and questions regarding the topological and metric properties of IFS dimension spectra.
We investigate subsets of a multifractal decomposition of the limit set of a conformal graph directed Markov system, which is constructed from the Cayley graph of a free group with at least two generators. The subsets we consider are parametrised by a normal subgroup $N$ of the free group and mimic the radial limit set of a Kleinian group. Our main results show that, regarding the Hausdorff dimension of these sets, various results for Kleinian groups can be generalised. Namely, under certain natural symmetry assumptions on the multifractal decomposition, we prove that, for a subset parametrise…
David maps are generalizations of classical planar quasiconformal maps for which the dilatation is allowed to tend to infinity in a controlled fashion. In this note we examine how these maps distort Hausdorff dimension. We show \vs {enumerate} [$\bullet$] Given $α$ and $β$ in $[0,2]$, there exists a David map $ϕ:\CC \to \CC$ and a compact set $Λ$ such that $\Hdim Λ=α$ and $\Hdim ϕ(Λ)=β$. \vs [$\bullet$] There exists a David map $ϕ:\CC \to \CC$ such that the Jordan curve $Γ=ϕ(\Sen)$ satisfies $\Hdim Γ=2$.\vs {enumerate} One should contrast the first statement with the fact that quasiconformal m…
On measures and semiconjugacies for affine interval exchange transformations
本文研究双曲周期型仿射区间交换变换的不变测度与共形测度的豪斯多夫维数，给出中心稳定型条件下的精确公式，并推导周期情形下半共轭的最高 Hölder 指数显式公式。
Exact dimensions of the prime continued fraction Cantor set
本文研究了素数Cantor集的Hausdorff维数和packing维数，证明了其Hausdorff测度不能有限正，但在数论猜想下其packing测度关于特定维数函数正且有限。
In this paper we compute the dimension of a class of dynamically defined non-conformal sets. Let $X\subseteq\mathbb{T}^2$ denote a Bedford-McMullen set and $T:X\to X$ the natural expanding toral endomorphism which leaves $X$ invariant. For an open set $U\subset X$ we let X_U={x\in X : T^k(x)\not\in U \text{for all}k}. We investigate the box and Hausdorff dimensions of $X_U$ for both a fixed Markov hole and also when $U$ is a shrinking metric ball. We show that the box dimension is controlled by the escape rate of the measure of maximal entropy through $U$, while the Hausdorff dimension depends…
We refine the multifractal formalism for the local dimension of a Gibbs measure $μ$ supported on the attractor $Λ$ of a conformal iterated functions system on the real line. Namely, for given $α\in \mathbb{R}$, we establish the formalism for the Hausdorff dimension of level sets of points $x\inΛ$ for which the $μ$-measure of a ball of radius $r_{n}$ centered at $x$ obeys a power law $r_{n}{}^α$, for a sequence $r_{n}\rightarrow0$. This allows us to investigate the Hölder regularity of various fractal functions, such as distribution functions and conjugacy maps associated with conformal iterate…
One of the key challenges in the dimension theory of smooth dynamical systems is in establishing whether or not the Hausdorff, lower and upper box dimensions coincide for invariant sets. For sets invariant under conformal dynamics, these three dimensions always coincide. On the other hand, considerable attention has been given to examples of sets invariant under non-conformal dynamics whose Hausdorff and box dimensions do not coincide. These constructions exploit the fact that the Hausdorff and box dimensions quantify size in fundamentally different ways, the former in terms of covers by sets …
Complex Diophantine Approximations and Cusp Excursions
通过Hurwitz映射的统一方法，研究复丢番图逼近渐近逼近率的Hausdorff维数谱与Bianchi轨道流形上尖点游荡的渐近平均游荡时间的Hausdorff维数谱。
We show that Bowen's equation, which characterises the Hausdorff dimension of certain sets in terms of the topological pressure of an expanding conformal map, applies in greater generality than has been heretofore established. In particular, we consider an arbitrary subset Z of a compact metric space and require only that the lower Lyapunov exponents be positive on Z, together with a tempered contraction condition. Among other things, this allows us to compute the dimension spectrum for Lyapunov exponents for maps with parabolic periodic points, and to relate the Hausdorff dimension to the top…
For Cantor circle Julia sets of hyperbolic rational maps, we prove that they are quasisymmetrically equivalent to standard Cantor circles (i.e., connected components are round circles). This gives a quasisymmetric uniformization of all Cantor circle Julia sets of hyperbolic rational maps. By analyzing the combinatorial information of the rational maps whose Julia sets are Cantor circles, we give a computational formula of the number of the Cantor circle hyperbolic components in the moduli space of rational maps for any fixed degree. We calculate the Hausdorff dimensions of the Julia sets which…
Degenerate perturbations of infinite graph-directed iterated function systems
研究非强连通无穷图导引迭代函数系统，给出极限集Hausdorff维数估计与Bowen型公式，并证明扰动系统维数收敛性，推广了有限图结果。
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