Questions tagged [hausdorff-dimension]

Questions about dimensions of possibly highly irregular or "rough" sets, Hausdorff–Besicovitch dimension and related concepts such as box-counting or Minkowski–Bouligand dimension. To be used with [tag:fractals] or [tag:dimension-theory].

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the intermediate property for topological Hausdorff dimension

According to page $889$, Theorem $3.6$ of A new fractal dimension..., the topological Hausdorff dimension of a subset $X$ of $\mathbb{R}^n$(or a subset of a separable metric space), can be defined as ...
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40 views

Last step in proof of countable stability of Hausdorff dimension

In part of Kenneth Falconer's proof of the countable stability of Hausdorff dimension on p. 49, sect 3.2 of Fractal Geometry, I understand him to say that $$\dim_H \bigcup_{i=1}^{\infty}F_i\leq \sup_{...
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22 views

Hausdorff dimension of sum of sets

Assume $0<r_0<n$. Are there sets $A,B\subseteq \mathbb{R}^n$, such that the Hasudorff dimension of $A,B$ are zero, But $\dim_H(A+B)=r_0$? When $r_0$ is integer, I have found(By attention to page ...
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25 views

Hausdorff dimension of product omega-languages

In this question I will refer to Hausdorff dimension and measure of regular $\omega$-languages as defined here. Hereby I give a quick recap of the definitions before diving into the question. Recap An ...
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70 views

Hausdorff dimension of a dense orbit in the Lorenz attractor

If I am not mistaken, then: the Lorenz attractor $\mathcal{A}$ has Hausdorff dimension $\dim_H(\mathcal{A}) > 2$, and the Lorenz attractor $\mathcal{A}$ contains a dense orbit $\mathcal{O}$, i.e. ...
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28 views

How can I measure or approximate the fractal dimension of the Barnsley Fern?

I would like to calculate a fractal dimension of the Barnsley Fern, but I am not sure what method may I use, nor even what fractal dimension I should use for this fractal. I know in this post it's ...
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64 views

Upper bound on the exact Hausdorff measure

Let the $s$-dimensional Hausdorff measure for a Borel set $F \subset \mathbb{R}^d$ be denoted as follows: $$\mathcal{H}^s(F) := \lim_{\delta \to 0} \mathcal{H}^s_\delta(F) := \lim_{\delta \to 0} \inf ...
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84 views

Growth rate of the Hausdorff measure

Let the $s$-dimensional Hausdorff measure for a Borel set $F \subset \mathbb{R}^d$ be denoted as follows: $$\mathcal{H}^s(F) := \lim_{\delta \to 0} \mathcal{H}^s_\delta(F) := \lim_{\delta \to 0} \inf ...
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46 views

Hausdorff dimension of the relative complement of a set contained in a $G_\delta$

We have the following result: Every set is contained in a $G_\delta$ set of the same Hausdorff dimension I was wondering how tight can this inclusion be made, complement-wise. Is true that: Let ...
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18 views

Hausdorff decomposition of a measure

Let $\mu$ be a probability measure on $(\mathbb R^n,\mathcal B^n)$. Is it true that $\mu$ can be decomposed as a countable sum of measures which are only distributed on sets of a fixed Hausdorff ...
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44 views

Hausdorff dimension of Sierpinski triangle less than log3/log2

Hi there I am struggling to understand the Hausdorff dimension of the Sierpinski triangle $S$. Below is I did to prove that $\alpha=\frac{\log 3}{\log 2}$, what should I do for $\alpha \le \frac{\log ...
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55 views

How to define a notion of Hausdorff homeomorphism?

A separable metric space is called fractal if its Hausdorff and topological dimensions are different. The Hausdorff dimension is not invariant by homeomorphism (see this post). Question: How to ...
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18 views

Whitney theorem in C^1 manifold

It's very well known the embedding theorem for smooth-manifolds. I want to know if is there any equivalent result for Lipschitz manifold and $C^1$-manifolds. In particular, I want that the embedding ...
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21 views

Can the fractal dimension of a coastline be less than 1?

I am currently investigating the fractal dimension of the coastline of the island the Palm Jumeirah, including the crescent. Using the Hausdorff method I have reached an answer of 0.879. This is less ...
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14 views

Find dim$_H$ $(\phi(S^1))$.

Assume that $φ:S^1\to \mathbb{R}^2$ is a map such that for some $L\ge 1$ and $0< s <1$ we have $\frac{1}{L}|x−y|^s≤|φ(x)−φ(y)|≤L|x−y|^s$ for all $x,y \in S^1$. Find dim$_H$$(\phi(S^1))$. So I ...
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30 views

Hausdorff dimension of a continuous curve at least 1.

This seems like it should be a simple problem but I am not sure how to solve it. For a non-constant continuous curve $\gamma : [0,1] \to \mathbb{R}^d$ show that the Hausdorff dimension of the image is ...
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31 views

Box-counting dimension (Minkowski Dimension) of {1, 1/4, 1/9, 1/16,…}

I want to determine the box-counting dimension (Minkowski Dimension) of the set $S=\{\frac{1}{n^2} \ | \ n \in \mathbb{Z}^+\}$. My first attempt was to first define my $\delta_n$. What I did was ...
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17 views

How do you find the Hausdorff dimension of a fractal curve given by an equation?

For example I was playing around in Desmos and came across the following fractal:$$y=\sum_{i=0}^{\infty}\cos(b^ix)b^{-i}$$So I'm curious to know what the fractal dimension is as a function of $b$ ( ...
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33 views

A set with infinity Hausdorff measure, but Hausdorff dimension $\frac{\log2}{\log3}$

I am going through the text 'Fractal Geometry: Mathematical Foundations and Applications' and came the following exercise: Give a set $B\subset\mathbb{R}$ that has Hausdorff dimension $s=\frac{\log2}{...
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30 views

Hausdorff Measure of Linear orthogonal map

Question: Suppose $T:\mathbb{R}^n\rightarrow\mathbb{R}^m$ is a linear orthogonal map $(n<m)$. Prove that $$\mathcal{H}^s(T(A))=\mathcal{H}^s(A)$$ for all $0\le s$. $\\$ ${\color{red}{\text{What ...
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23 views

Show for a $\gamma-$holder map $\mathcal{H}^{s/\gamma}(f(A))\le MH^s(A)$

Suppose $f: \mathbb{R}^n\rightarrow \mathbb{R}^m$ is a $\gamma-$Holder continuous map $(0<\gamma\le1)$, that is:} $$|f(x)-f(y)|\le C |x-y|^\gamma, \ \ \ \ \text{for some }C>0, \text{ and all }x,...
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13 views

box dimension of set

Let F consist of those numbers in $[0, 1]$ whose decimal expansions do not contain the digit $5$. Find $dim_BF$.
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How do I calculate the Hausdorff dimension of a self-affine fractal (like the Barnsley Fern)?

The fractal I am concerned with has an infinite number of self-affine copies of itself, and all scaled to different dimensions. And all but one of them are rotated too. I know this may sound way more '...
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43 views

Hausdorff dimension for bounded set

Let $F$ be the set of of numbers $x\in [0,1]$ with base 3 expansions $0.a_1a_2...$ for which there exists an integer k such that $a_i\neq 1$ for all $i\geq k$. Find the Hausdorff dimension of $F$. ...
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24 views

How do you calculate Hausdorff-dimension of real world objects when Hausdorff-dimension needn't to be equal to other dimensions?

In general Hausdorff-dimension is never larger as Minkowski / Box-counting dimension. For some sets like self-similar sets both dimension coincidence. In literature you can read that the Hausdorff-...
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19 views

Hausdorff dimension of graph of composition of functions

Given two functions $f,g$, is there a reasonable bound of the Hausdorff dimension of the graph of $f\circ g$ given the Hausdorff dimensions of the graphs of $f$ and $g$? For example, does it hold that ...
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18 views

Are there interesting general properties held by sets with small Hausdorff dimension?

Let us focus on the plane. I am wondering whether the sets with small (but non zero) Hausdorff dimension share some interesting properties. Let $\epsilon\in(0,1)$ and assume that a set $F\subset \...
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Hausdorff measure finite and positive

Let $A\subseteq [0,1]^d$ be a Borel measurable set. Let $d_H(A)$ be the Hausdorff dimension of $A$. Let $\mathcal{H}^{d_H(A)}$ be the Hausdorff measure w.r.t to the dimension of $A$. My question is, ...
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56 views

Complement of strictly less than (n-1)-dimensional set is connected?

Let $n\in \mathbb{N}$ and $A\subseteq \mathbb{R}^n$ be a subset such that its Hausdorff dimension is strictly less than $n-1$. Is it then true that $\mathbb{R}^n\setminus A$ is (path-)connected? ...