# 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).

201 questions
Filter by
Sorted by
Tagged with
399 views

### How to plot this fractal

I'm a graphic designer and my client has asked me to use this fractal in a design that I'm working on. As you can see, it's not a very good copy, so I'm trying to see if I can generate a high-...
31 views

### Hausdorff dimension of cookie-cutter sets using topological pressure

I'm reading the proof of Theorem 5.3 from "Techniques in Fractal Geometry" by Kenneth Falconer - the theorem wich gives the Hausdorff dimension of a cookie-cutter set using topological ...
79 views

56 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-...
246 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 ...
79 views

183 views

108 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 ...
97 views

64 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?
171 views

195 views

### Lipschitz property of holonomies fails when stable leaves $W^s(x)$ inside the leaves $W^{ss}(x)$

Let $M$ be compact manifold. suppose $f:M\rightarrow M$ is $C^{2}$. There is a continuous splitting of the tangent bundle $TM=E^{ss}+E^{s}+E^{u}$ invariant under the derivative $Df$ of the ...
268 views

### continuity entropy with respect gibbs measures

Let $X=\{0,1\}^{\mathbb{N}}$. For simplicity I consider Bernoulli measures on $X$ only. Let $f:X\to \mathbb{R}$ be Holder continuous. The measure $\mu$ is a Gibbs measure with potential $f$ if there ...
114 views

### Is each Peano continuum a topological fractal?

Problem. Is each Peano continuum a topological fractal? A compact Hausdorff space $X$ is a topological fractal if $X=\bigcup_{i=1}^n f_i(X)$ for some continuous maps $f_1,\dots,f_n:X\to X$ such that ...
110 views

### How many two-dimensional space filling Hilbert-like curves are there?

I'm interested in filling 2d square with space filling, non-self-intersecting, locality preserving, self-similar curves, like Hilbert curve. I found interesting work concerning three dimensional case ...
406 views

155 views

### Examples of probability measures with `fake' decay

To be concise, I am wondering whether there are natural examples of probability measures $\mu$ compactly supported on the real line which satisfy $\mu(I) \lesssim l_n^\alpha$ for all intervals $I$ ...
604 views

### Hausdorff dimension of the graph of an increasing function

Let $f$ be a continuous, strictly increasing function from $[0,1]$ to itself with $f(0)=0, f(1)=1$. Let $\Gamma_f$ denote its graph. What can be said about the Hausdorff dimension of $\Gamma_f$? In ...
79 views

### On the convergence problem of box counting for the Rössler attractor

So 5 month ago i posted this Question: Rössler attractor, Convergence of box counting to estimate the fractal dimension. Since then i have assumed, that the rate of convergence of $n(ɛ,n)$ (sum ...
Let $s\in (0, 1)$ be a real number. Let $E\subset [0, 1]$ be a Borel set whose Hausdorff dimension is given by $s$. Assume that $\mathcal{H}^s(E)=+\infty$, that is, the $s$-dimensional Hausdorff ...