Stable intersection of middle-$α$ Cantor sets

On the measure of products from the middle-third Cantor set

We prove upper and lower bounds for the Lebesgue measure of the set of products $xy$ with $x$ and $y$ in the middle-third Cantor set. Our method is inspired by Athreya, Reznick and Tyson, but a different subdivision of the Cantor set provides a more rapidly converging approximation formula.

Diffusion on middle-$ξ$ Cantor sets

In this paper, we study $C^ζ$-calculus on generalized Cantor sets, which have self-similar properties and fractional dimensions that exceed their topological dimensions. Functions with fractal support are not differentiable or integrable in terms of standard calculus, so we must involve local fractional derivatives. We have generalized the $C^ζ$-calculus on the generalized Cantor sets known as middle-$ξ$ Cantor sets. We have suggested a calculus on the middle-$ξ$ Cantor sets for different values of $ξ$ with $0<ξ<1$. Differential equations on the middle-$ξ$ Cantor sets have been solved, and we …

Affine embeddings and intersections of Cantor sets

Let $E, F\subset \R^d$ be two self-similar sets. Under mild conditions, we show that $F$ can be $C^1$-embedded into $E$ if and only if it can be affinely embedded into $E$; furthermore if $F$ can not be affinely embedded into $E$, then the Hausdorff dimension of the intersection $E\cap f(F)$ is strictly less than that of $F$ for any $C^1$-diffeomorphism $f$ on $\R^d$. Under certain circumstances, we prove the logarithmic commensurability between the contraction ratios of $E$ and $F$ if $F$ can be affinely embedded into $E$. As an application, we show that $\dim_HE\cap f(F)<\min\{\dim_HE, \dim_…

Stable intersections of regular conformal Cantor sets with large Hausdorff dimensions

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…

Approximating elements of the middle third Cantor set with dyadic rationals

Let $C$ be the middle third Cantor set and $μ$ be the $\frac{\log 2}{\log 3}$-dimensional Hausdorff measure restricted to $C$. In this paper we study approximations of elements of $C$ by dyadic rationals. Our main result implies that for $μ$ almost every $x\in C$ we have $$\#\left\{1\leq n\leq N:\left|x-\frac{p}{2^n}\right| \leq \frac{1}{n^{0.01}\cdot 2^{n}}\textrm{ for some }p\in\mathbb{N}\right\}\sim 2\sum_{n=1}^{N}n^{-0.01}.$$ This improves upon a recent result of Allen, Chow, and Yu which gives a sub-logarithmic improvement over the trivial approximation rate.

$\mathbb{C}^2$ 自同构同宿分岔的 Cantor 集稳定相交与持久切触的正密度

Stable intersections of Cantor sets and positive density of persistent tangencies for homoclinic bifurcations of automorphisms of $\mathbb{C}^2$

本文研究 $\mathbb{C}^2$ 自同构族的通有同宿切触，证明了若稳定与不稳定流形局部相交 Cantor 集的线性化版本稳定相交，则持久切触参数集在分岔点处具有正密度。

关于康托集交集与随机矩阵乘积的交集

On the intersection of Cantor sets and products of random matrices

本文提出新的工具，计算连续 stationary 测度下的 Lyapunov 指数和 Hausdorff 维数，并应用于单个数字禁止情况下的维数计算。

幂和，以及中间三分之一 Cantor 集元素的乘积

Sums of powers, and products of elements of the middle third Cantor set

本文提出一种新的初等动力学技术，统一处理中间三分之一 Cantor 集元素的幂和与幂积的测度及开区间问题，克服了已有方法在幂次增大时的困难。

Intersections of middle-$α$ Cantor sets with a fixed translation

For $λ\in(0,1/3]$ let $C_λ$ be the middle-$(1-2λ)$ Cantor set in $\mathbb R$. Given $t\in[-1,1]$, excluding the trivial case we show that \[ Λ(t):=\left\{λ\in(0,1/3]: C_λ\cap(C_λ+t)\ne\emptyset\right\} \] is a topological Cantor set with zero Lebesgue measure and full Hausdorff dimension. In particular, we calculate the local dimension of $Λ(t)$, which reveals a dimensional variation principle. Furthermore, for any $β\in[0,1]$ we show that the level set \[ Λ_β(t):=\left\{λ\inΛ(t): \dim_H(C_λ\cap(C_λ+t))=\dim_P(C_λ\cap(C_λ+t))=β\frac{\log 2}{-\log λ}\right\} \] has equal Hausdorff and packing d…

Iterated functions and the Cantor set in one dimension

In this paper we consider the long-term behavior of points in ${\mathbb R}$ under iterations of continuous functions. We show that, given any Cantor set $Λ^*$ embedded in ${\mathbb R}$, there exists a continuous function $F^*:{\mathbb R}\to{\mathbb R}$ such that the points that are bounded under iterations of $F^*$ are just those points in $Λ^*$. In the course of this, we find a striking similarity between the way in which we construct the Cantor middle-thirds set, and the way in which we find the points bounded under iterations of certain continuous functions.

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…

高维Cantor集I：稳定相交的判据

Cantor sets in higher dimension I: Criterion for stable intersections

本文在高维空间中研究Cantor集的稳定相交判据，并提出构造显式稳定相交Cantor集的方法，适用于Hausdorff维度极小的复杂情形。

Interiors of continuous images of the middle-third Cantor set

Let $C$ be the middle-third Cantor set, and $f$ a continuous function defined on an open set $U\subset \mathbb{R}^{2}$. Denote the image \begin{equation*} f_{U}(C,C)=\{f(x,y):(x,y)\in (C\times C)\cap U\}. \end{equation*} If $\partial _{x}f$, $\partial _{y}f$ are continuous on $U,$ and there is a point $(x_{0},y_{0})\in (C\times C)\cap U$ such that \begin{equation*} 1<\left\vert \frac{\partial _{x}f|_{(x_{0},y_{0})}}{\partial _{y}f|_{(x_{0},y_{0})}}\right\vert <3\text{ or }1<\left\vert \frac{\partial _{y}f|_{(x_{0},y_{0})}}{\partial _{x}f|_{(x_{0},y_{0})}}\right\vert <3, \end{equation*} then $f…

Products of two Cantor sets

We consider products of two Cantor sets, and obtain the optimal estimates in terms of their thickness that guarantee that their product is an interval. This problem is motivated by the fact that the spectrum of the Labyrinth model, which is a two dimensional quasicrystal model, is given by a product of two Cantor sets. We also discuss the connection with the question on the structure of intersections of two Cantor sets, which was considered by many authors previously.

A Complex Gap Lemma

Inspired by the work of Newhouse in one real variable, we introduce a relevant notion of thickness for dynamical Cantor sets of the plane associated to a holomorphic IFS. Our main result is a complex version of Newhouse's Gap Lemma : we show that under some assumptions, if the product t(K)t(L) of the thicknesses of two Cantor sets K and L is larger than 1, then K and L have non empty intersection. Since in addition this thickness varies continuously, this gives a criterion to get a robust intersection between two Cantor sets in the plane.

Intersections of homogeneous Cantor sets and beta-expansions

Let $Γ_{β,N}$ be the $N$-part homogeneous Cantor set with $β\in(1/(2N-1),1/N)$. Any string $(j_\ell)_{\ell=1}^\N$ with $j_\ell\in\{0,\pm 1,...,\pm(N-1)\}$ such that $t=\sum_{\ell=1}^\N j_\ellβ^{\ell-1}(1-β)/(N-1)$ is called a code of $t$. Let $\mathcal{U}_{β,\pm N}$ be the set of $t\in[-1,1]$ having a unique code, and let $\mathcal{S}_{β,\pm N}$ be the set of $t\in\mathcal{U}_{β,\pm N}$ which make the intersection $Γ_{β,N}\cap(Γ_{β,N}+t)$ a self-similar set. We characterize the set $\mathcal{U}_{β,\pm N}$ in a geometrical and algebraical way, and give a sufficient and necessary condition for $…

Cantor集上的平均Fourier估计与二进逼近

Averaged Fourier Estimates and Dyadic Approximation on the Cantor set

本文通过平均Fourier估计和二进逼近方法，研究了Cantor集上二进逼近的零一律，改进了Velani猜想的已知结果。

On the arithmetic sums of Cantor sets

Let $C_\la$ and $C_\ga$ be two affine Cantor sets in $\mathbb{R}$ with similarity dimensions $d_\la$ and $d_\ga$, respectively. We define an analog of the Bandt-Graf condition for self-similar systems and use it to give necessary and sufficient conditions for having $\Ha^{d_\la+d_\ga}(C_\la + C_\ga)>0$ where $C_\la + C_\ga$ denotes the arithmetic sum of the sets. We use this result to analyze the orthogonal projection properties of sets of the form $C_\la \times C_\ga$. We prove that for Lebesgue almost all directions $θ$ for which the projection is not one-to-one, the projection has zero $(d_…

Rare events for Cantor target sets

We study the existence of limiting laws of rare events corresponding to the entrance of the orbits on certain target sets in the phase space. The limiting laws are obtained when the target sets shrink to a Cantor set of zero Lebesgue measure. We consider both the presence and absence of clustering, which is detected by the Extremal Index, which turns out to be very useful to identify the compatibility between the dynamics and the fractal structure of the limiting Cantor set. The computation of the Extremal Index is connected to the box dimension of the intersection between the Cantor set and i…

Intersections of multiplicative translates of 3-adic Cantor sets

Motivated by a question of Erdős, this paper considers questions concerning the discrete dynamical system on the 3-adic integers given by multiplication by 2. Let the 3-adic Cantor set consist of all 3-adic integers whose expansions use only the digits 0 and 1. The exception set is the set of 3-adic integers whose forward orbits under this action intersects the 3-adic Cantor set infinitely many times. It has been shown that this set has Hausdorff dimension 0. Approaches to upper bounds on the Hausdorff dimensions of these sets leads to study of intersections of multiplicative translates of Can…

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。

A note on dyadic approximation in Cantor's set

We consider the convergence theory for dyadic approximation in the middle-third Cantor set, $K$, for approximation functions of the form $ψ_τ(n) = n^{-τ}$ ($τ\ge 0$). In particular, we show that for values of $τ$ beyond a certain threshold we have that almost no point in $K$ is dyadically $ψ_τ$-well approximable with respect to the natural probability measure on $K$. This refines a previous result in this direction obtained by the first, third, and fourth named authors (arXiv, 2020).

关于两个仿射Cantor集的和

Topological structure of the sum of two affine Cantor sets

该文研究仿射Cantor集和集的维数与结构性质，证明其盒维数与Hausdorff维数相等且对应Hausdorff测度有限，给出维数和不超过1时的零测结果，并刻画双递增映射仿射Cantor集和集的五种generic结构。

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…

Multiplication on uniform $λ$-Cantor sets

Let $C$ be the middle-third Cantor set. Define $C*C=\{x*y:x,y\in C\}$, where $*=+,-,\cdot,÷$ (when $*=÷$, we assume $y\neq0$). Steinhaus \cite{HS} proved in 1917 that \[ C-C=[-1,1], C+C=[0,2]. \] In 2019, Athreya, Reznick and Tyson \cite{Tyson} proved that \[ C÷C=\bigcup_{n=-\infty}^{\infty}\left[ 3^{-n}\dfrac{2}{3},3^{-n}\dfrac {3}{2}\right] . \] In this paper, we give a description of the topological structure and Lebesgue measure of $C\cdot C$. We indeed obtain corresponding results on the uniform $λ$-Cantor sets.

Ratio geometry, rigidity and the scenery process for hyperbolic Cantor sets

Given a $C^{1+γ}$ hyperbolic Cantor set $C$, we study the sequence $C_{n,x}$ of Cantor subsets which nest down toward a point $x$ in $C$. We show that $C_{n,x}$ is asymptotically equal to an ergodic Cantor set valued process. The values of this process, called limit sets, are indexed by a Hölder continuous set-valued function defined on D. Sullivan's dual Cantor set. We show the limit sets are themselves $C^{k+γ}, C^\infty$ or $C^ω$ hyperbolic Cantor sets, with the highest degree of smoothness which occurs in the $C^{1+γ}$ conjugacy class of $C$. The proof of this leads to the following rigidi…

三分康托集上二进逼近的度量结果

Metric results for dyadic approximation on the middle-third Cantor set

研究三分康托集上二进逼近的度量性质，通过建立傅里叶系数估计，改进了Velani猜想的已知范围，将零测部分阈值降至约1.429，满测部分阈值升至约0.052。

On the self similarity of generalized Cantor sets

We consider the self-similar structure of the class of generalized Cantor sets $$Γ_{\mathcal{D}}=\Big\{\sum_{n=1}^\infty d_nβ^{n}: d_n\in D_n, n\ge 1\Big\},$$ where $0<β<1$ and $D_n, n\ge 1,$ are nonempty and finite subsets of $\mathbb{Z}$. We give a necessary and sufficient condition for $Γ_{\mathcal{D}}$ to be a homogeneously generated self similar set. An application to the self-similarity of intersections of generalized Cantor sets will be given.

Unique expansions and intersections of Cantor sets

To each $α\in(1/3,1/2)$ we associate the Cantor set $$Γ_α:=\Big\{\sum_{i=1}^{\infty}ε_{i}α^i: ε_i\in\{0,1\},\,i\geq 1\Big\}.$$ In this paper we consider the intersection $Γ_α\cap (Γ_α+ t)$ for any translation $t\in\mathbb{R}$. We pay special attention to those $t$ with a unique $\{-1,0,1\}$ $α$-expansion, and study the set $$D_α:=\{\dim_H(Γ_α\cap (Γ_α+ t)):t \textrm{ has a unique }\{-1,0,1\}\,α\textrm{-expansion}\}.$$ We prove that there exists a transcendental number $α_{KL}\approx 0.39433\ldots$ such that: $D_α$ is finite for $α\in(α_{KL},1/2),$ $D_{α_{KL}}$ is infinitely countable, and $D_α…

由复基数展开导出的 Cantor 集的交集

Intersections of Cantor Sets Derived from Complex Radix Expansions

本文研究复基数 IFS Cantor 集平移交集的维数稠密性，放宽数字间距与大小限制，并将结论推广到 Hausdorff、packing 维数及交集自相似性分类。

Intersections of multiplicative translates of $3$-adic Cantor sets II: two infinite families

This paper continues the study of the structure of finite intersections of general multiplicative translates $\mathcal{C}(M_1,\ldots,M_n)=\frac{1}{M_1}Σ_{3,\bar{2}}\cap\cdots\cap\frac{1}{M_n}Σ_{3,\bar{2}}$ for integers $1\leq M_1<\cdots<M_n$, where $Σ_{3,\bar{2}}$ denotes the $3$-adic Cantor set. This study was motivated by questions concerning the discrete dynamical system on the $3$-adic integers $\mathbb{Z}_3$ given by multiplication by $2$. The exceptional set $\mathcal{E}(\mathbb{Z}_3)$ is defined to be the set of all elements of $\mathbb{Z}_3$ whose forward orbits under this action inter…

How likely can a point be in different Cantor sets

Let $m\in\mathbb N_{\ge 2}$, and let $\mathcal K=\{K_λ: λ\in(0, 1/m]\}$ be a class of Cantor sets, where $K_λ=\{\sum_{i=1}^\infty d_iλ^i: d_i\in\{0,1,\ldots, m-1\}, i\ge 1\}$. We investigate in this paper the likelyhood of a fixed point in the Cantor sets of $\mathcal K$. More precisely, for a fixed point $x\in(0,1)$ we consider the parameter set $Λ(x)=\{λ\in(0,1/m]: x\in K_λ\}$, and show that $Λ(x)$ is a topological Cantor set having zero Lebesgue measure and full Hausdorff dimension. Furthermore, by constructing a sequence of Cantor subsets with large thickness in $Λ(x)$ we prove that the in…

康托集上受扰连续映射相图的灵活性与通用性

Flexibility versus genericity of phase diagrams of perturbed continuous maps on the Cantor set

该研究针对有限字母表上受随机噪声扰动的连续映射动力系统，刻画了使系统具有唯一测度的噪声参数集合的结构，并证明通有情形下该集合为开区间左开右闭的(0,1]，揭示了相图的灵活性与通有性质。

A survey on Newhouse thickness, fractal intersections and patterns

In this article, we introduce a notion of size for sets called thickness that can be used to guarantee that two Cantor sets intersect (the Gap Lemma), and show a connection among Thickness, Schmidt Games and Patterns. We work mostly in the real line, but we also introduce the topic in higher dimensions.

The Minkowski sum of linear Cantor sets

Let $C$ be the classical middle third Cantor set. It is well known that $C+C = [0,2]$ (Steinhaus, 1917). (Here $+$ denotes the Minkowski sum.) Let $U$ be the set of $z \in [0,2]$ which have a unique representation as $z = x + y$ with $x, y \in C$ (the set of uniqueness). It isn't difficult to show that $\dim_H U = \log(2) / \log(3)$ and $U$ essentially looks like $2C$. Assuming $0,n-1 \in A \subset \{0,1,\dots,n-1\}$, define $C_A = C_{A,n}$ as the linear Cantor set which the attractor of the iterated function system \[ \{ x \mapsto (x + a) / n: a \in A \}. \] We consider various properties of …

Random Walks, Spectral Gaps, and Khintchine's Theorem on Fractals

This work addresses problems on simultaneous Diophantine approximation on fractals, motivated by a long standing problem of Mahler regarding Cantor's middle $1/3$ set. We obtain the first instances where a complete analogue of Khintchine's Theorem holds for fractal measures. Our results apply to fractals which are self-similar by a system of rational similarities of $\mathbb{R}^d$ (for any $d\geq 1$) and have sufficiently small Hausdorff co-dimension. A concrete example of such measures in the context of Mahler's problem is the Hausdorff measure on the "middle $1/5$ Cantor set"; i.e. the set o…

关于同构自相似集与其平移在R^n中的交集以及在Z^n中的乘法不变性公式的提出

On the intersections of homogeneous self-similar sets with their translates in $\mathbb{R}^{n}$ and a formulation of multiplicative invariance in $\mathbb{Z}^{n}$

本文研究了自仿集在R^n中的交集性质，提出了乘法不变性的定义，并探讨了其与n维环面不变集的关系。

$L^q$ dimensions of self-similar measures, and applications: a survey

We present a self-contained proof of a formula for the $L^q$ dimensions of self-similar measures on the real line under exponential separation (up to the proof of an inverse theorem for the $L^q$ norm of convolutions). This is a special case of a more general result of the author from [Shmerkin, Pablo. On Furstenberg's intersection conjecture, self-similar measures, and the $L^q$ norms of convolutions. Ann. of Math., 2019], and one of the goals of this survey is to present the ideas in a simpler, but important, setting. We also review some applications of the main result to the study of Bernou…

Spatially independent martingales, intersections, and applications

We define a class of random measures, spatially independent martingales, which we view as a natural generalisation of the canonical random discrete set, and which includes as special cases many variants of fractal percolation and Poissonian cut-outs. We pair the random measures with deterministic families of parametrised measures $\{η_t\}_t$, and show that under some natural checkable conditions, a.s. the total measure of the intersections is Hölder continuous as a function of $t$. This continuity phenomenon turns out to underpin a large amount of geometric information about these measures, al…