About $C^{1}$-minimality of the hyperbolic Cantor sets

Anosov 与部分双曲微分同胚的强叶状结构的极小性

Minimality of Strong Foliations of Anosov and Partially Hyperbolic Diffeomorphisms

本文研究了部分双曲微分同胚扩张不变叶状结构的极小性，引入了在$C^1$扰动下稳健的s-横截性，并证明了在弱扩张条件下强不稳定叶状结构的稳健极小性及其在环面和唯一Gibbs测度上的应用。

Stable minimality of expanding foliations

We prove that generically in $\text{Diff}^{1}_{m}(M)$, if an expanding $f$-invariant foliation $W$ of dimension $u$ is minimal and there is a periodic point of unstable index $u$, the foliation is stably minimal. By this we mean there is a $C^{1}$-neighborhood $\mathcal{U}$ of $f$ such that for all $C^{2}$-diffeomorphisms $g\in \mathcal{U}$, the $g$-invariant analytic continuation of $W$ is minimal. In particular, all such $g$ are topologically mixing. Moreover, all such $g$ have a hyperbolic ergodic component of the volume measure $m$ which is essentially dense. This component is, in fact, Be…

Semiconjugacies, pinched Cantor bouquets and hyperbolic orbifolds

Let f be a transcendental entire map that is subhyperbolic, i.e., the intersection of the Fatou set F(f) and the postsingular set P(f) is compact and the intersection of the Julia set J(f) and P(f) is finite. Assume that no asymptotic value of f belongs to J(f) and that the local degree of f at all points in J(f) is bounded by some finite constant. We prove that there is a hyperbolic map g (of the form g(z)=f(bz) for some complex number b) with connected Fatou set such that f and g are semiconjugate on their Julia sets. Furthermore, we show that this semiconjugacy is a conjugacy when restricte…

Nested Cantor sets

We give sufficient conditions for two Cantor sets of the line to be nested for a positive set of translation parameters. This problem occurs in diophantine approximations. It also occurs as a toy model of the parameter selection for non-uniformly hyperbolic attractors of the plane. For natural Cantors sets, we show that this condition is optimal.

Dynamics of certain smooth one-dimensional mappings IV: Asymptotic geometry of Cantor sets

We study hyperbolic mappings depending on a parameter $\varepsilon $. Each of them has an invariant Cantor set. As $\varepsilon $ tends to zero, the mapping approaches the boundary of hyperbolicity. We analyze the asymptotics of the gap geometry and the scaling function geometry of the invariant Cantor set as $\varepsilon $ goes to zero. For example, in the quadratic case, we show that all the gaps close uniformly with speed $\sqrt {\varepsilon}$. There is a limiting scaling function of the limiting mapping and this scaling function has dense jump discontinuities because the limiting mapping i…

具有Collet-Eckmann点和无限多个汇的野集：稳定性和共存

Non-Hyperbolic Chaotic Dynamics: Renormalization and More

本文研究了二维系统中野集与非均匀混沌超限性的共存现象，并提出了一种广义重正化方案。

Quasisymmetric uniformization and Hausdorff dimensions of Cantor circle Julia sets

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…

A variational principle for systems with nonuniformly hyperbolic behavior with applications to the dimension theory

Let $f$ be a $C^{1+α}$ nonuniformly hyperbolic diffeomorphism. We use a a nonadditive version of the topological pressure of a class of admissible, possibly noncontinuous potentials $P^*(Φ)$ to prove the following variational equation: $P^*(Φ) = \sup_{Ω\in {\mathcal H}}P^*(f|Ω,Φ)$ supremum taken over the set ${\mathcal H}$ of basic subsets in $M$. As a consequence we find a lower bound for the Cantor dimension of the stable and unstable Cantor sets of a non trivial conformal nonuniformly hyperbolic isolated sets.

Robust minimality of strong foliations for DA diffeomorphisms: $cu$-volume expansion and new examples

Let $f$ be a $C^2$ partially hyperbolic diffeomorphisms of ${\mathbb T}^3$ (not necessarily volume preserving or transitive) isotopic to a linear Anosov diffeomorphism $A$ with eigenvalues $$λ_{s}<1<λ_{c}<λ_{u}.$$ Under the assumption that the set $$\{x: \,\mid\log \det(Tf\mid_{E^{cu}(x)})\mid \leq \log λ_{u} \}$$ has zero volume inside any unstable leaf of $f$ where $E^{cu} = E^c\oplus E^u$ is the center unstable bundle, we prove that the stable foliation of $f$ is $C^1$ robustly minimal, i.e., the stable foliation of any diffeomorphism $C^1$ sufficiently close to $f$ is minimal. In particula…

Minimality of the action on the universal circle of uniform foliations

Given a uniform foliation by Gromov hyperbolic leaves on a $3$-manifold, we show that the action of the fundamental group on the universal circle is minimal and transitive on pairs of different points. We also prove two other results: we prove that general uniform Reebless foliations are $\mathbb{R}$-covered and we give a new description of the universal circle of $\mathbb{R}$-covered foliations with Gromov hyperbolic leaves in terms of the JSJ decomposition of $M$.

Global topology of hyperbolic components I: Cantor circle case

The hyperbolic components in the moduli space ${M}_d$ of degree $d\geq2$ rational maps are mysterious and fundamental topological objects. For those in the connectedness locus, they are known to be the finite quotients of the Euclidean space $\mathbb{R}^{4d-4}$. In this paper, we study the hyperbolic components in the disconnectedness locus and with minimal complexity: those in the Cantor circle locus. We show that each of them is a finite quotient of the space $\mathbb{R}^{4d-4-n}\times\mathbb{T}^{n}$, where $n$ is determined by the dynamics. The proof relates Riemann surface theory (Abel's T…

From Cantor to Semi-hyperbolic Parameter along External Rays

For the quadratic family $f_{c}(z) = z^2+c$ with $c$ in the exterior of the Mandelbrot set, it is known that every point in the Julia set moves holomorphically. Let $\hat{c}$ be a semi-hyperbolic parameter in the boundary of the Mandelbrot set. In this paper we prove that for each $z = z(c)$ in the Julia set, the derivative $dz(c)/dc$ is uniformly $O(1/\sqrt{|c-\hat{c}|})$ when $c$ belongs to a parameter ray that lands on $\hat{c}$. We also characterize the degeneration of the dynamics along the parameter ray.

Stable intersection of Cantor sets in higher dimension and robust homoclinic tangency of the largest codimension

In this paper, we construct (a) a pair of two regular Cantor sets in higher dimension which exhibits $C^1$-stable intersection and (b) a hyperbolic basic set which exhibits $C^2$-robust homoclinic tangency of the largest codimension for any higher dimensional manifold, using blenders. The former implies that an analog of Moreira's theorem on Cantor sets in the real line does not hold in higher dimension. The latter solves a question posed by Barrientos and A.Raibekas.

Rational maps whose Julia sets are Cantor circles

In this paper, we give a family of rational maps whose Julia sets are Cantor circles and show that every rational map whose Julia set is a Cantor set of circles must be topologically conjugate to one map in this family on their corresponding Julia sets. In particular, we give the specific expressions of some rational maps whose Julia sets are Cantor circles, but they are not topologically conjugate to any McMullen maps on their Julia sets. Moreover, some non-hyperbolic rational maps whose Julia sets are Cantor circles are also constructed.

No invariant line fields on Cantor Julia sets

In this paper, we prove that a rational map with a Cantor Julia set carries no invariant line fields on its Julia set. It follows that a structurally stable rational map with a Cantor Julia set is hyperbolic.

篱笆上的动力学

Dynamics on fences

本文提出一种将康托集上的动力学映射推广到相应篱笆空间的动力学构造，从而统一研究多种篱笆状空间的动力学特性，并建立其拓扑结构与动力行为之间的新联系。

动力学定义的 Cantor 集的 Hausdorff 维数新界

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。

Cantor Spectrum for Schrödinger Operators with Potentials arising from Generalized Skew-shifts

We consider continuous $SL(2,\mathbb{R})$-cocycles over a strictly ergodic homeomorphism which fibers over an almost periodic dynamical system (generalized skew-shifts). We prove that any cocycle which is not uniformly hyperbolic can be approximated by one which is conjugate to an $SO(2,\mathbb{R})$-cocycle. Using this, we show that if a cocycle's homotopy class does not display a certain obstruction to uniform hyperbolicity, then it can be $C^0$-perturbed to become uniformly hyperbolic. For cocycles arising from Schrödinger operators, the obstruction vanishes and we conclude that uniform hype…

Hyperbolic dimension of Julia sets of meromorphic maps with logarithmic tracts

We prove that for meromorphic maps with logarithmic tracts (e.g. entire or meromorphic maps with a finite number of poles from class $\mathcal B$), the Julia set contains a compact invariant hyperbolic Cantor set of Hausdorff dimension greater than 1. Hence, the hyperbolic dimension of the Julia set is greater than 1.

Remarks on the dynamics of the horocycle flow for homogeneous foliations by hyperbolic surfaces

This article is a first step towards the understanding of the dynamics of the horocycle flow on foliated manifolds by hyperbolic surfaces. This is motivated by a question formulated by M. Martinez and A. Verjovsky on the minimality of this flow assuming that the natural affine foliation is minimal too. We have tried to offer a simple presentation, which allows us to update and shed light on the classical theorem proved by G. A. Hedlund in 1936 on the minimality of the horocycle flow on compact hyperbolic surfaces. Firstly, we extend this result to the product of PSL(2,R) and a Lie group G, whi…

Quasisymmetric geometry of the Cantor circles as the Julia sets of rational maps

We give three families of parabolic rational maps and show that every Cantor set of circles as the Julia set of a non-hyperbolic rational map must be quasisymmetrically equivalent to the Julia set of one map in these families for suitable parameters. Combining a result obtained before, we give a complete classification of the Cantor circles Julia sets in the sense of quasisymmetric equivalence. Moreover, we study the regularity of the components of the Cantor circles Julia sets and establish a sufficient and necessary condition when a component of a Cantor circles Julia set is a quasicircle.

Non-Smooth Bifurcations of Uniformly Hyperbolic Invariant Manifolds in Skew Product Systems: Rigorous Results

In this paper we study the anti-integrable limit scenario of skew-product systems. We consider a generalization of such systems based on the Frenkel-Kontorova model, and prove the existence of orbits with any fibered rotation number in systems of both one and two degrees of freedom. In particular, our results also apply to two dimensional maps with degenerate potentials (vanishing second derivative), so extending the results of existence of Cantori for more general twist maps. We also prove that under certain mild regularity conditions on the potential the structure of the orbits is of Cantor …

Hausdorff dimension of Cantor intersections and robust heterodimensional cycles for heterochaos horseshoe maps

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…

On Dynamics of λ+ tan z^2

This article discusses some topological properties of the dynamical plane ($z$-plane) of the holomorphic family of meromorphic maps $λ+ \tan z^2$ for $ λ\in \mathbb C$. In the dynamical plane, we prove that there is no Herman ring, and the Julia set is a Cantor set for the maps when the parameter is in the unbounded hyperbolic component contained in the four quadrants in the complex plane. Julia set is connected for the maps when the parameters are in other hyperbolic components of the parameter plane.

亏格 $g$ 康托集与相关 Julia 集

Genus $g$ Cantor sets and germane Julia sets

本文利用康托集的亏格研究三维空间中一致拟正则映射的 Julia 集，首次构造了每点局部亏格为 $g$ 的亏格 $g$ 康托集并实现为 Julia 集，同时证明了双曲情形下康托 Julia 集亏格有限且局部亏格在稠密子集上出现。

完美核与动力学：从Bass-Serre理论到双曲群

Perfect kernel and dynamics: from Bass-Serre theory to hyperbolic groups

本文通过引入新方法研究可数群子群空间的Cantor-Bendixson分解与完美核，确定了许多新群的完美核和秩，建立了共轭作用的拓扑传递性条件，并由此得到类A中大量新的群例，涵盖直角Artin群、极限群、小消去群、随机群及虚拟紧特殊群等。

二维Cantor集II：稳定交集的最优维度约束

Cantor sets in higher dimensions II: Optimal dimension constraint for stable intersections

本文证明了对于正则Cantor集，上盒维数之和小于d时可被小变形分离，且此约束是最佳的。通过构造具有任意小厚度的正则Cantor集对，展示了C^{1+α}稳定交集的条件。

具有 Cantor bouquet Julia 集的整函数

Entire functions with Cantor bouquet Julia sets

本文研究超越整函数的Cantor bouquet Julia集，证明了criniferous性质与发丝结构的等价性，给出基于吸收集的新刻画，并在几何假设下证明head-start条件既是充分的也是必要的。

λ tan z^2 切族映射的动力学

Dynamics of the family $λ$ tangent $z^2$

本文研究亚纯映射族 λ tan z^2 的动力学平面拓扑性质，证明在含原点的双曲分支中无 Herman 环且 Julia 集为 Cantor 集，在其他双曲分支中 Julia 集连通。

关于圆周微分同胚的随机游走与群作用的维数理论

Dimension theory of group actions by circle diffeomorphisms II: Minimal sets

本文研究圆周微分同胚随机游走与群作用的不变集维数，证明例外极小集的Hausdorff维数小于1，并在含重数为k+1的不动点时大于k/(k+1)，从而将Fuchs群经典结果推广到非线性情形。

沿内部射线从双曲参数到抛物参数

From Hyperbolic to Parabolic Parameters along Internal Rays

本文研究二次族沿内部射线趋近抛物参数时 Julia 集点的全纯运动，给出一致导数估计，并证明指数仅依赖花瓣数的一致单侧 Hölder 连续性。

On uniformly disconnected Julia sets

It is well-known that the Julia set of a hyperbolic rational map is quasisymmetrically equivalent to the standard Cantor set. Using the uniformization theorem of David and Semmes, this result comes down to the fact that such a Julia set is both uniformly perfect and uniformly disconnected. We study the analogous question for Julia sets of UQR maps in $\mathbb{S}^n$, for $n\geq 2$. Introducing hyperbolic UQR maps, we show that the Julia set of such a map is uniformly disconnected if it is totally disconnected. Moreover, we show that if $E$ is a compact, uniformly perfect and uniformly disconnec…

The Octagonal PET II: The Topology of the Limit Sets

This is a sequel to my paper "The Octagonal PET I: Renormalization and Hyperbolic Symmetry". In this paper we use the renormalization scheme found in the first paper to classify the limit sets of the systems according to their topology. The main result is that the limit set is either a finite forest or a Cantor set, with an explicit description of which cases occur for which parameters. In one special case, the limit set is a disjoint union of 2 arcs if and only if the continued fraction expansion of the parameter has the form [a0:a1:a2:a3...] with a_k even for every odd k.

仿射评估映射、维数群与公平测度

Affine Evaluation Maps, Dimension Groups, and Fair measures

本文研究仿射评估映射（AEM），构造了与良态AEM对应的单维数群，利用相关定理证明其可由极小康托尔同胚实现，并将该框架推广到可数可和群的极小康托尔作用。

Julia 集在 Mandelbrot 集中拟共形地出现

Julia sets appear quasiconformally in the Mandelbrot set

本文证明在小 Mandelbrot 集的抛物型或 Misiurewicz 参数附近可拟共形地看到 Cantor Julia 集及其嵌套装饰结构，并由此证明半双曲但非 Misiurewicz 且非双曲的参数在 Mandelbrot 集边界中稠密且 Hausdorff 维数为 2。

Dynamical properties and structure of Julia sets of postcritically bounded polynomial semigroups

We discuss the dynamic and structural properties of polynomial semigroups, a natural extension of iteration theory to random (walk) dynamics, where the semigroup $G$ of complex polynomials (under the operation of composition of functions) is such that there exists a bounded set in the plane which contains any finite critical value of any map $g \in G$. In general, the Julia set of such a semigroup $G$ may be disconnected, and each Fatou component of such $G$ is either simply connected or doubly connected (\cite{Su01,Su9}). In this paper, we show that for any two distinct Fatou components of ce…