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 maps preserve sets of Hausdorff dimension 0 and 2. The second statement provides an example of a Jordan curve with Hausdorff dimension 2 which is (quasi)conformally removable.
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.
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…
By employing the recurrence method worked out in `Estimating the Hausdorff measure by recurrence', we provide effective lower estimates of the proper--dimensional Hausdorff measure of minimal sets of circle homeomorphisms that are not conjugate to any rotation.
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.
We prove the existence of rational maps whose Julia sets are Sierpiński carpets having positive area. Such rational maps can be constructed such that they either contain a Cremer fixed point, a Siegel disk or are infinitely renormalizable. We also construct some Sierpiński carpet Julia sets with zero area but with Hausdorff dimension two. Moreover, for any given number $s\in(1,2)$, we prove the existence of Sierpiński carpet Julia sets having Hausdorff dimension exactly $s$.
In the moduli space $\mathcal{M}_d$ of degree $d$ rational maps, the bifurcation locus is the support of a closed $(1,1)$ positive current $T_\bif$ which is called the bifurcation current. This current gives rise to a measure $μ_\bif:=(T_\bif)^{2d-2}$ whose support is the seat of strong bifurcations. Our main result says that $\supp(μ_\bif)$ has maximal Hausdorff dimension $2(2d-2)$. As a consequence, the set of degree $d$ rational maps having $2d-2$ distinct neutral cycles is dense in a set of full Hausdorff dimension.
In \cite{B} Bourgain proves that Sarnak's disjointness conjecture holds for a certain class of Three-interval exchange maps. In the present paper we slightly improve the Diophantine condition of Bourgain and estimate the constants in the proof. We further show, that the new parameter set has positive, but not full Hausdorff dimension. This, in particular, implies that the Lebesgue measure of this set is zero.
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.
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 …
We give examples of infinitely renormalizable quadratic polynomials $F_c: z\maps to z^2+c$ with stationary combinatorics whose Julia sets have Hausdorff dimension arbitrar y close to 1. The combinatorics of the renormalization involved is close to the Chebyshev one . The argument is based upon a new tool, a ``Recursive Quadratic Estimate'' for the Poincaré series of an infinitely re normalizable map.
We study the dynamics of the map $x$ to $dx$ (mod 1) on the unit circle. We characterize the invariant finite subsets of this map which are called cycles and are graded by their degrees. By looking at the combinatorial properties of the base-d expansion of the elements in the cycles, we prove a conjecture of Curt McMullen that the Hausdorff dimension of the closure of degree-m cycles is equal to log $m$ / log $d$.
Sun Daochun and Yang Lo have shown that many results of the Fatou-Julia iteration theory of rational functions extend to quasiregular self-maps of the Riemann sphere for which the degree exceeds the dilatation. We show that in this context, in contrast to the case of rational functions, the Julia set may have Hausdorff dimension zero. On the other hand, we exhibit a gauge function depending on the degree and the dilatation such that the Hausdorff measure with respect to this gauge function is always positive, but may be finite.
We study bifurcations of invariant graphs in skew product dynamical systems driven by hyperbolic surface maps T like Anosov surface diffeomorphisms or baker maps and with one-dimensional concave fibre maps under multiplicative forcing when the forcing is scaled by a parameter r=e^{-t}. For a range of parameters two invariant graphs (a trivial and a non-trivial one) coexist, and we use thermodynamic formalism to characterize the parameter dependence of the Hausdorff and packing dimension of the set of points where both graphs coincide. As a corollary we characterize the parameter dependence of …
As a model to provide a hands-on, elementary understanding of chaotic dynamics in dimension three, we introduce a $C^2$-open set of diffeomorphisms of $\mathbb R^3$ having two horseshoes with different dimensions of instability. We prove that: the unstable set of one horseshoe and the stable set of the other are of Hausdorff dimension nearly $2$ whose cross sections are Cantor sets; the intersection of the unstable and stable sets contains a fractal set of Hausdorff dimension nearly $1$. As a corollary we detect $C^2$-robust heterodimensional cycles. Our proof employs the theory of normally hy…
We discuss on some families of skew product maps on a square. For a kind of skew product maps with coupled-expanding property, we estimate Hausdorff dimension of its attractor. And we prove that there exists an ergodic measure with full Hausdorff dimension for this kind of skew product maps.
Piecewise contracting maps on the interval: Hausdorff dimension, entropy and attractors
Escuela de Ingeniería, Facultad de Ingeniería y Empresa – Universidad Católica Silva Henríquez（天主教希尔瓦·恩里克兹大学工程学院、工商学院）；Department of Engineering Mathematics – University of Bristol（布里斯托大学工程数学系）
Let $f:\bar\bold C\to\bar\bold C$ be a rational map on the Riemann sphere , such that for every $f$-critical point $c\in J$ which forward trajectory does not contain any other critical point, $|(f^n)'(f(c))|$ grows exponentially fast (Collet--Eckmann condition), there are no parabolic periodic points, and else such that Julia set is not the whole sphere. Then smooth (Riemann) measure of the Julia set is 0. For $f$ satisfying additionally Masato Tsujii's condition that the average distance of $f^n(c)$ from the set of critical points is not too small, we prove that Hausdorff dimension of Julia s…
In this paper, we introduce a new entropy-like invariant, named Hausdorff metric entropy, for finitely generated semigroups acting on compact metric spaces from a set-valued view and study its properties. We establish the relation between Hausdorff metric entropy and topological entropy of a semigroup defined by Biś. Some examples with positive or zero Hausdorff metric entropy are given. Moreover, some notions of chaos are also well generalized for finitely generated semigroups from a set-valued view.
We investigate the set $I$ of parameters $κ$ for which the singular value of $z\mapsto e^z+κ$ converges to $\infty$. The set $I$ consists of uncountably many parameter rays, plus landing points of some of these rays. We show that the parameter rays have Hausdorff dimension 1, while the ray endpoints in $I$ alone have dimension 2. Analogous results were known for dynamical planes of exponential maps; our result shows that this also holds in parameter space.
In this paper we deal with the following family of exponential maps $(f_λ:z\mapsto λ(e^z-1))_{λ\in [1,+\infty)}$. Denoting $d(λ)$ the hyperbolic dimension of $f_λ$. It is known that the function $λ\mapsto d(λ)$ is real analytic in $(1,+\infty)$, and that it is continuous in $[1,+\infty)$. In this paper we prove that this map is C$^1$ on $[1,+\infty)$, with $d'(1^+)=0$. Moreover, depending on the value of $d(1)$, we give estimates of the speed of convergence towards 0.
We construct suitable metrics for two classes of topological dynamical systems (linear maps on the torus and non-invertible expansive maps on compact spaces) in order to get a lower bound for topological entropy in terms of the resulting Hausdorff dimensions and Lipschitz constants. This reverses an old inequality of Dai, Zhou, and Gheng and leads to a short proof of a well-known theorem on expansive mappings. It also suggests a new invariant of topological conjugacy for dynamical systems on compact metric spaces.
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 $2d/(d+1)$.
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.
We show that attractors are semicontinuous for closed relations on compact Hausdorff spaces. Semicontinuity is what guarantees that small changes to a system do not result in massive growth of certain features, notably attractors. That is, there is a certain preservation of structure. When it comes to flows, semiflows, and maps, it is well established that attractors are semicontinuous. In [2], relations were established as a way to generalize maps, and a formal definition of attractors was established. Relations (in the dynamical systems sense) represent discrete time systems, which may lack …
Let $σ_{K}: \sum_{K}\rightarrow \sum_{K}$ be a shift map. For an interval $[p,q]\subset[0,1]$, let $D_{σ_{K}}([p,q])$ denote the set of pairs for which the density spectrum of the $ε$-approach time set equals $[p,q]$ when $ε$ is small and $E_{σ_{K}}([p,q])$ the set of pairs for which the density spectrum of the $ε$-approach time set converges to $[p,q]$ when $ε\rightarrow 0^+$. Then $\dim_{H} D_{σ_{K}}([p,q])=\dim_{H} E_{σ_{K}}([p,q])=2-q$. Moreover, $\mathscr{H}^{2-q}(E_{σ_{K}}([p,q]))=1$ when $q=0$ and $\mathscr{H}^{2-q}(E_{σ_{K}}([p,q]))=+\infty$ when $q>0$. Meanwhile, $\mathscr{H}^{2-q}(D_…
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.
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…
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 the case of hyperbolic basic sets $Λ$ of saddle type for holomorphic maps $f: \mathbb P^2\mathbb C \to \mathbb P^2\mathbb C$. We study equilibrium measures $μ_ϕ$ associated to a class of Hölder potentials $ϕ$ on $Λ$, and find the measures $μ_ϕ$ of iterates of arbitrary Bowen balls. Estimates for the pointwise dimension $δ_{μ_ϕ}$ of $μ_ϕ$ that involve Lyapunov exponents and a correction term are found, and also a formula for the Hausdorff dimension of $μ_ϕ$ in the case when the preimage counting function is constant on $Λ$. For terminal/minimal saddle sets we prove that an invariant…
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 i…
We study the multifractal analysis for smooth dynamical systems in dimension one. It is characterized the Hausdorff dimension of the level set obtained from the Birkhoff averages of a continuous function by the local dimensions of hyperbolic measures for a topologically mixing $C^2$ map modelled by an abstract dynamical system. A characterization which corresponds to above is also given for the ergodic basins of invariant probability measures. And it is shown that the complement of the set of quasi-regular points carries full Hausdorff dimension.
For each irrational $α\in[0,1)$ we construct a continuous function $f\: [0,1)\to \R$ such that the corresponding cylindrical transformation $[0,1)\times\R \ni (x,t) \mapsto (x+α, t+ f(x)) \in [0,1)\times\R$ is transitive and the Hausdorff dimension of the set of points whose orbits are discrete is 2. Such cylindrical transformations are shown to display a certain chaotic behaviour of Devaney-like type.
We study differentiability of topological conjugacies between expanding piecewise $C^{1+ε}$ interval maps. If these conjugacies are not $C^1$, then they have zero derivative almost everywhere. We obtain the result that in this case the Hausdorff dimension of the set of points for which the derivative of the conjugacy does not exist lies strictly between zero and one. Using multifractal analysis and thermodynamic formalism, we show that this Hausdorff dimension is explicitly determined by the Lyapunov spectrum. Moreover, we show that these results give rise to a "rigidity dichotomy" for the typ…
In this paper, we study the Hausdorff dimension of the generalized intrinsic level set with respect to the given ergodic meausre in a class of non-uniformly hyperbolic interval maps with finitely many branches.
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…
We use the upper and lower potential functions and Bowen's formula estimating the Hausdorff dimension of the limit set of a regular semigroup generated by finitely many $C^{1+α}$-contracting mappings. This result is an application of the geometric distortion lemma in the first paper at this series.
For a positive measure set of nonuniformly expanding quadratic maps on the interval we effect a multifractal formalism, i.e., decompose the phase space into level sets of time averages of a given observable and consider the associated {\it Birkhoff spectrum} which encodes this decomposition. We derive a formula which relates the Hausdorff dimension of level sets to entropies and Lyapunov exponents of invariant probability measures, and then use this formula to show that the spectrum is continuous. In order to estimate the Hausdorff dimension from above, one has to "see" sufficiently many point…
We consider the dynamics of expanding semigroups generated by finitely many rational maps on the Riemann sphere. We show that for an analytic family of such semigroups, the Bowen parameter function is real-analytic and plurisubharmonic. Combining this with a result obtained by the first author, we show that if for each semigroup of such an analytic family of expanding semigroups satisfies the open set condition, then the function of the Hausdorff dimension of the Julia set is real-analytic and plurisubharmonic. Moreover, we provide an extensive collection of classes of examples of analytic fam…
In this paper, we introduce the notion of asymptotic self-similar sets on general doubling metric spaces by extending the notion of self-similar sets, and determine their Hausdorff dimensions, which gives an extension of Balogh and Rohner 's result. This is carried out by introducing the notions of almost similarity maps and asymptotic similarity systems. These notions have an advantage of making geometric constructions possible. Actually, as an application, we determined the Hausdorff dimension of general Sierpinski gaskets on complete surfaces constructed by a geometric way in a natural mann…
We construct Feigenbaum quadratic polynomials whose Julia sets have positive Lebesgue measure. They provide first examples of rational maps for which the hyperbolic dimension is different from the Hausdorff dimension of the Julia set. The corresponding set of parameters has positive Hausdorff dimension.
For a strongly dissipative Hénon-like map at the first bifurcation parameter at which the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set, we effect a multifractal analysis, i.e., decompose the set of non wandering points on the unstable manifold into level sets of an unstable Lyapunov exponent, and give a partial description of the Lyapunov spectrum which encodes this decomposition. We derive a formula for the Hausdorff dimension of the level sets in terms of the entropy and unstable Lyapunov exponent of invariant probability measures, and show the conti…
