# Questions tagged [fractals]

Fractals deal with special sets that exhibit complicated patterns in every scale. Fractal sets usually have a Hausdorff dimension different from its topological dimension. Examples include Julia sets, the Sierpinski triangle, the Cantor set. Fractals naturally appear in dynamical system, such as iterations in the complex plane, or as strange attractors to continuous dynamical systems, (see Lorentz attractor).

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### A question about box dimension and Hölder condition

It is known that if a real continuous function $f(x)$ satisfies a local $\alpha$-Hölder condition on a closed interval $[a,b]$, the box dimension of the graph of $f(x)$ on $[a,b]$ will be not greater ...
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### Is there a description of the points of the Cantor set on which the Cantor function is differentiable?

Let $C$ be the usual ternary cantor set, and $f$ the Cantor function, or Devil’s staircase associated to it. We know that $f$ is differentiable a.e., and on every point of the complement $C^c$, the ...
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### Hemispherical space filling hilbert curve

First question here, sorry for any posting infractions. I need to create/find/buy a hemispherical space-filling Hilbert(or similar) curve. something similar to Cube hilbert but only filling a ...
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### Is this result on the set of differentiability of the distance function to the fat cantor set new?

Definitions: Let $C \subset [0, 1]$ be a fat Cantor set, for parameter $0 < r < 1/3$. Thus intervals of width $r^n$ are removed from the middle of the previous intervals at each step. For the ...
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### Closed set with full box dimension and non-full Hausdorff dimension

Tried posting this on math.SE first but didn't get any responses. Is there an example of a closed set $E \subseteq [0,1]$ such that $\dim_B(E) = 1$ and $\dim_H(E) < 1$? Here $\dim_B$ is box (...
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Many years ago I found an inequality that directly controlled whether a point $\displaystyle c$ belongs or does not belong to the Mandelbrot set. Roughly, it was something like this: If $\displaystyle ... 1answer 88 views ### Box counting dimension and Besov spaces on$\mathbb R^2$I found a lemma in this paper of Constantin and Wu, stated with no proof: Lemma 3.2. Let$b=\chi_{D}$be the characteristic function of a bounded domain$D\subset\mathbb R^2$whose boundary has box-... 1answer 300 views ### Convex Julia sets Consider the classical Julia set$J_f$associated with$f(z)=z^2+c$. Since$J_c$is completely invariant, we know that$f^{-1}(J_f) \subseteq J_f$. Now, let$H_f$be the convex hull of$J_f$. Is it ... 1answer 82 views ### Entropy spectrum is not concave Let$T:[0, 1]\rightarrow [0, 1]$be map such that$T(x)=4x(1-x)$. For any$\alpha \in \mathbb{R}$, we define the level set as follows $$F(\alpha)=\{x\in [0,1]: \lim_{n\rightarrow \infty}\frac{1}{n}\... 0answers 115 views ### Can this number be interpreted as a fractal dimension? Under Goldbach's conjecture, let's denote for a large enough integer n by r_{0}(n) the quantity \inf\{r>0,(n-r,n+r)\in\mathbb{P}^2\} and by k_{0}(n):=\pi(n+r_{0}(n))-\pi(n+r_{0}(n)). Let's ... 2answers 491 views ### Algorithm for computing external angles for the Mandelbrot set Let M be the Mandelbrot set: there exists a unique series$$ \psi(z) := z + \sum_{m=0}^{+\infty} b_m z^{-m} = z - \frac{1}{2} + \frac{1}{8} z^{-1} - \frac{1}{4} z^{-2} + \cdots $$which defines a ... 3answers 2k views ### How to add two numbers from a group theoretic perspective? It is known that adding two numbers and looking at the carrying operation has a link with cocycles in group theory. (https://www.jstor.org/stable/3072368?origin=crossref) When we add two numbers by ... 2answers 904 views ### A set whose Hausdorff dimension gradually changes? Can there be a set whose Hausdorff dimension gradually changes? For instance, a set of real numbers contained in an interval, whose Hausdorff dimension is 0 at the beginning and 1 closer to the end, ... 0answers 81 views ### Hausdorff dimension between (1,2) Is there a number c \in (1,2) for which there exist some interesting geometric property/properties which hold for every set of Hausdorff dimension in (1,c) and does not hold for any set of ... 0answers 188 views ### Is Domineering on any finite approximation of the Sierpinski Carpet always a second-player win? The game of Domineering can be played on any board consisting of some subset of \mathbb{Z} \times \mathbb{Z}. In particular, consider the boards K_n generated by iterating the following inductive ... 0answers 94 views ### How do sets with unit fractional Hausdorff measure of dimension >1 look like? Triggered by the recent question How can we not know the measure of the Sierpiński triangle? I would like to ask: Let s>1 and s not be an integer. How to construct a set A with \mathfrak{H}^... 1answer 185 views ### Set operations over iterated function systems An iterated function system (IFS) is a finite set of contraction mappings on a complete metric space. Symbolically,$$\{f_i :X \to X \mid i=1,2,..n \},\quad n \in \mathbb{N}$$is an IFS if each ... 1answer 753 views ### How can we not know the s-measure of the Sierpiński triangle? I'm preparing a presentation that would enable high-school level students to grasp that the (self-similarity) dimension of an object needs not be an integer. The first example we look at is the ... 2answers 709 views ### Who proved that the Mandelbrot set's Julia sets are locally connected? I'd be greatly interested in a reference to the respective article. Was it Douady? Julia? Hubbard? Fatou? Bonus question: Who gave the proof that can be found in the Orsay notes? EDIT: The question ... 0answers 36 views ### Numerically Evaluate the limit of the solution of a functional equation I need to evaluate the limit of f(x) as x\to0, where the function f solves the following equation:$$ f(x)=\left\{ \begin{array}{ll} g(x) & \text{if } x\geq \frac{1}{2};\\ \frac{1}{2} f(\... 1answer 254 views ### Wavefront set of characteristic function of rough set It is a standard exercise to show that if$X\subseteq\mathbb{R}^n$has smooth boundary, then the characteristic function$1_X$has wavefront set $$\{(x,\xi)\in\partial X\times\mathbb{R}^n\setminus\{0\}... 1answer 650 views ### The Koch snow flake, Holder exponents of conformal mappings The Koch snow flake K is a domain of \mathbb{C}, complex plane. Though, I do not state the precise definition, you can see the picture in wikipedia Koch snow flake. The Koch snow flake K is a ... 1answer 162 views ### Does fractallity depend on the Riemannian metric? Edit: According to comment of Andre Henriques we revise the question: In this question a fractal is a metric space whose topological and Hausdorff dimensions are different. So we would like that ... 0answers 229 views ### Hausdorff dimension and von Neumann dimension There are two subjects in which non-integral dimensions appear: fractal geometry: consider the well-known Hausdorff dimension of fractals. von Neumann algebra: consider a type {\rm II_1} ... 1answer 76 views ### Hausdorff outer measure is finite if \sum_{j = 1}^m |f(x_i)-f(x_{i-1})|^s \le c [closed] Let f:[0,1] \to \mathbb{R}, G = graph(f). If \sum_{j = 1}^m |f(x_i)-f(x_{i-1})|^s \le c for all partitions 0 = x_0< \ldots < x_m = 1 then H^s(G) < \infty What technique can I use to ... 0answers 267 views ### Covering lemmas in Hochman's ''On self-similar sets with overlaps and inverse theorems for entropy'' I am confused about the covering lemmas in the captioned work and really hope to get some ideas here. Firstly it is lemma 3.7. (Image of Lemma 3.7) (for convenience here is the lemma of this lemma (... 0answers 117 views ### Fourier coeffients of Cantor measure For 0<\theta<\frac{1}{2}, denote by \mu_\theta the uniform Cantor measure with dissection ratio \theta. It is not hard to show that the Fourier–Stieltjes transform of \mu_\theta is$$ \... 1answer 114 views ### Pointless characterization relating between a fractal and its code space Given an hyperbolic IFS$(X,\{f_i:i=1,\ldots,N\})$and denoting its code space by$\Sigma_N = \{1,\ldots,N\}^{\mathbb{N}}$and the generated fractal set by$\mathcal{A}$. There is a continuous and ... 0answers 99 views ### Evaluating this limit in Fourier analysis In my research work related to Fourier transform of the standard Cantor measure, I came across the following elementary problem: For$k\geq 1$, let $$S_k=\sum_{m=0}^{3^k-1}\,\,\,\prod_{j=1}^\infty \... 1answer 93 views ### Box dimension as the critical value of the fractal content Let M \subseteq \mathbb{R}^n be bounded and N_{\epsilon}(M) the minimum number of 'squares' of side \epsilon with center in M necessary to cover M. The box dimension of M is then defined as \... 1answer 167 views ### Formal justification of the Chaos game in the Sierpinski triangle I want to justify why the Chaos game works to produce Sierpinski triangle. I use a theorem taken from Massopust Interpolation and Approximation with Splines and Fractals. Suppose that (X,d) is a ... 2answers 203 views ### Box dimension of the graph of an increasing function This Hausdorff dimension of the graph of an increasing function shows that: Let f be a continuous, strictly increasing function from [0,1] to itself with f(0)=0, f(1)=1. Then dim_H \; G = ... 1answer 138 views ### Failure of Falconer distance problem in one dimension I am told that the Falconer distance conjecture fails trivially in one dimension, but I really cannot find any reference for that. Precisely, I am asking the following question: For a compact set E\... 0answers 44 views ### Quasilinear elliptic problem on fractal domain Consider the following quasilinear elliptic equation$$\nabla_x (A(x,u(x),\nabla_x u(x))) + f(u,x) = 0 $$on a bounded domain \Omega, augmented with homogeneous Dirichlet boundary data:$$u|_{\... 0answers 72 views ### Set with modified lower box counting dimension strictly less than Hausdorff dimension Please, can someone give (as simple as possible) example of the set for which modified lower box counting dimension is strictly smaller than Hausdorff dimension? 0answers 180 views ### Has this self-similar sequence the ratio$(\sqrt2+1)^2$? This is inspired by a math.SE question, where an infinite sequence of pairwise distinct natural numbers$a_1=1, a_2, a_3, ...$has been defined as follows:$a_n$is the smallest number such that$s_n:=...
Let $(X,d)$ be a complete metric space, $$M^1 := \{\mu: \mu \mbox{ is a Borel regular measures having bounded support and } \mu(X) = 1\},$$ and BC(X) := \{f : f:X\rightarrow \mathbb{R} \mbox{ is ...