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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29 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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1answer
35 views

Strict Convexity and Set of Failures of P.D. Hessian

Suppose $f: \mathbb{R}^n \to \mathbb{R}$ is $C^2$ and strictly convex. Consider the set of points in the domain for which the Hessian matrix of $f$ fails to be positive definite: $$S = \big \{x \in \...
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21 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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1answer
41 views

Fractal dimension of a fractal that is comprised of discrete objects

I'm trying to understand fractal dimension in the context of colloidal gels. But more on that later. I'm confused about a more fundamental thing, which I think relates to the discreteness of the ...
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1answer
35 views

Holomorphic maps preserve Hausdorff dimension.

In a paper I read there is the following claim: Let $f:\mathbb{C}\to \mathbb{C}$ be a non-constant entire transcendental function(essential singularity at infinity) and $A\subset \mathbb{C}$ a set in ...
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1answer
73 views

Sets of infinite Hausdorff dimension in a second countable metric space

I am wondering if there exists an example of a second countable metric space $X$ containing a set $A$ with infinite Hausdorff dimension.
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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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2answers
78 views

Solving for complex exponent

Background (though not necessary for the mechanics of what I'm asking): According to Lapidus' $\textit{Fractal Geometry, Complex Dimensions and Zeta Functions}$, the Cantor String consists of lengths $...
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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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1answer
161 views

Lower bound for upper $\pi/2$ angular density

This is exercise 2.3 in Falconer's book 'The Geometry of Fractal Sets'. Let $E\subset \mathbb{R}^n$ be an $\textit{s}$-set. That is, it is measurable for the s-dimensional Hausdorff measure $H^s$ and ...
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Hausdorff Dimension for Koch-like Fractal [duplicate]

I'm trying to define is a simple fractal: detach the middle third of a horizontal line, push it upwards by one third the length of the original segment and then add in vertical lines to keep it all ...
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28 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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1answer
36 views

The Hausdorff measure of the unit interval

I am trying to calculate the Hausdorff measure of the unit interval. Here's my attempt: Fix $\epsilon>0$. consider the open balls $B(x,\epsilon)$ with $x\in[0,1]$. How many can cover the unit ...
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1answer
58 views

Are there two sets $X$ and $Y$ such that the following inequality for box dimension holds

I am trying to find sets $X$ and $Y$ s.t. $\dim_B(X\bigcup{}Y)>\max\{\dim_B(X),\dim_B(Y)\}$. At first I thought taking $X=[0,1]$ and $Y=\{10+1/n^2:n\ge{}1\}$ but I don't think that works. Is this ...
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3answers
49 views

Finding the dimension of the sphere cube

If you take an $2r\times 2r\times 2r$ cube, and divide it to 27 equal cubes, and then remove all the "axis" cubes (all the cubes which are straight left, straight right, straight up, etc. from the ...
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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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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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2answers
48 views

Hausdorff distance and Hausdorff dimension

Let $X$ be a metric space, $A$, $B$ two compact subsets of $X$ such that the Hausdorff distance $dist_H$ between $A$ and $B$ is small: $$ dist_H(A, B) \leq \epsilon, $$ with $\epsilon > 0$. Does it ...
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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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83 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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1answer
23 views

Hausdorff dimension from above

Let $A_n$ be a set of Hausdorff dimension $1-\frac{1}{n}$ then, the set $$A=\cup_n A_n$$ Has Hausdorff dimension $1$ (nevertheless having $H_1(A)=0$). My question is: can we do the same thing from ...
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1answer
42 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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1answer
45 views

Hausdorff dimension and mass distribution

I'm struggling to understand part of a proof in Falconer's book on Fractal Geometry. It's Theorem 4.13 (a): Let $F$ be a subset of $\mathbb{R}^n$. If there is a mass distribution $\mu$ on $F$ with $...
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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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2answers
74 views

Hausdorff-dimension of non measurable sets?

The hausdorff-outer-measure is defined for all subsets of a metric space. The hausdorff measure is defined as the restriction to caratheodory measurable sets. I actually don't know how the set of ...
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1answer
43 views

Infinite Hausdorff dimension in discrete metric spaces

I was searching for a metric space that has infinite Hausdorff dimenion . I stumbled upon the example of $\mathbb{R}$ with discrete metric. $\mathbb{R}$ should then have infinite dimension but I ...
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Does not the existence of fractals with fractional Hausdorff dimension prove that there are cardinalities in between countable and continuum?

Is not it the case that the cardinality of fractals of dimension 0.00001 should be greater than countable but less than that of continuum?
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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

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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46 views

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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0answers
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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42 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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1answer
54 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? ...
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1answer
262 views

What is the fractal dimension/Hausdorff dimension of a Koch's snowflake?

I have found that the fractal dimension of a self-similar object is: $$\text{fractal dimension} = \frac{\log(\text{number of self-similar pieces})}{\log(\text{magnification factor})} $$ See here ...
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2answers
79 views

How big can the Hausdorff dimension of the closure of a smooth curve be?

Consider curves in $\mathbb{R}^n$. Smooth curves have Hausdorff dimension $1$. The closure of a smooth curve can have Hausdorff dimension $> 1$. (For example, a curve dense in a torus.) How big ...
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387 views

Hausdorff dimension of Hamiltonian orbit closure and symplectic leaves

Let $\dot{x} = \Pi \cdot \nabla H$ be a smooth Hamiltonian-Poisson system on $\mathbb{R}^n$. $H: \mathbb{R}^n \to \mathbb{R}$ is the Hamiltonian and $\Pi = (\Pi^{ij})$ is a skew-symmetric matrix of ...
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0answers
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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1answer
100 views

Hausdorff dimension of compact set

Let $(X, d)$ be a metric space and $S \subset X$. Let $\DeclareMathOperator{\diam}{diam}\diam(S)$ denote diameter of $S$, that is $\diam(S) = \sup \{ d(x, y) \colon \: x, y \in S \}$. Let $\delta > ...
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1answer
60 views

Hausdorff dimension of radial set

Let $A\subset [0,1]$ be a compact set of Hausdorff dimension $\alpha$. Let also $A_n:=\{x\in \mathbb{R}^n~:~ \|x\|_2 \in A\}$. Is it true that $\text{dim}_H(A_n)=n-1+\alpha$? I believe that this ...
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1answer
58 views

Similarity dimension and Hausdorff dimension

How do we prove the similarity dimension equals the Hausdorff dimension if the self-similar set satisfies the open set condition? Which article contains this proof?
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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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3answers
189 views

The Hausdorff dimension of the zero set of a real analytic function

Let $n>1$, and let $f:\mathbb{R}^n \to \mathbb{R}$ be a real-analytic function which is not identically zero. Does $\dim_{\mathcal H}(f^{-1}(0)) \le n-1$? here $\dim_{\mathcal H}$ refers to the ...
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1answer
79 views

Does the domain where a harmonic map is conformal has Hausdorff dimension smaller than one?

Let $\mathbb{D}^2$ be the closed unit disk, and let $f:\mathbb{D}^2 \to \mathbb{R}^2$ be a harmonic map, satisfying $\det df \neq 0$ almost everywhere on $\mathbb{D}^2$. Suppose also that $$U= \{ p \...
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1answer
56 views

Covering dimension of boundary of compact subset of $\mathbb{R}^n$

Let $X$ be a compact subset of $\mathbb{R}^n$, with the inherited Euclidean topology. Does it follow that $\dim_{cov}(\partial X)\leq n-1$? I would be happy to have a reference for that, if it is ...
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1answer
355 views

Fractal dimensional analysis?

I know how to use a ruler to approximate a length of an object (like a wire or a stick) in meters. I could also use the ruler to approximate a two dimensional area (like a table top or a parking lot) ...