# Tagged Questions

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 ...

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### Is there a survey of recent work relating to the Hausdorff dimension of sets defined through some restriction of digits?

I am familiar with the work of Helmut Cajar, but his book is thirty years old and it's clear that there has been substantial progress since then. I have been spending a lot of time looking through ...
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### Possible Hausdorff dimension of intersection of Besicovitch-Eggleston like sets

Let $b \geq 2$ be an integer and suppose that $v=(p_0,\cdots,p_{b-1})$ be a probability vector. Let $S_{b,v}$ be the set of real numbers whose $b$-ary expansion has the digit $k$ with relative ...
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### Relation between math and piano music

What, if any, is the relation between Cantor's function and Ligeti studio: Devil's Staircase?
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### Iterated function system on the plane

Let $r_1, r_2, r_3$ be three nonnegative real numbers with $r_1^2+r_2^2+r_3^2 <1$. Can you find three similitudes $f_1,f_2,f_3$ on $\mathbb{R}^2$ with similarity ratios $r_1,r_2,r_3$ resp. and a ...
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### Correspondence between fractal sets and trees

In Hillel Furstenberg's series lectures on ergodic theory in fractal geometry, he mentioned his search on finding a one-to-one correspondence between fractal sets and trees, however, I couldn't not ...
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### Analysis of the boundary of the Mandelbrot set

Motivation: The Mandlebrot set is a simply connected set with an infinitely complex boundary, but CAN one move from interior to the exterior of this topological space by just crossing over a finite ...
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### Is Gouvêa-Mazur's “Infinite Fern” a fractal?

[EDIT]: Following Qiaochu Yuan's comment, it is better to clarify that I do not know what the right definition of a fractal in the following question should be. But a nice answer might contain such a ...
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### Arithmetic products of Cantor sets.

Let $A,B\subseteq \mathbb{R}$ be two Cantor sets. What is known about the arithmetic product $AB=\lbrace ab|a\in A, b\in B\rbrace$? In particular, what is known in the case that the sets are self-...
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### Littlewood-Paley theory and norm estimation

In the paper "A Convolution Inequality Concerning Cantor-Lebesgue Measures", the Littlewood-Paley theory is used to estimate the norm of multiplier operator in Lemma 1. It is claimed that Lemma 2 is ...
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### What is known about the area of the symmetric Pythagorean tree?

What is known about the area of the symmetric Pythagorean tree? (Closed unit square as base, area of enclosed triangles not included.) The problem in calculating the area is that squares start to ...
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### L-systems and Sierpinski Triangle

I was just shocked when I saw these consecutive outcomes of an L-system converging to the Sierpinski triangle (shown in the picture below). I'm interested to know how could one arrange the rules of ...
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### Sierpinski Triangle and the Chaos Game

The chaos game is a way to construct (an approximation) of Sierpinski triangle. It's clear (using Thales' theorem!) that if we begin with a point on the sierpinski triangle, then we will never leave ...
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### Lecture on Fractals for Middle School Students

I'm going to have a one-hour lecture for middle school students next Monday. It will be about fractals. The students know virtually nothing about this subject. I'll show some fractal images and a few ...
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### Hausdorff dimension of non-recurrent walks

Preface: I am fairly new to the concept of Hausdorff dimension, so I don't know how interesting a question this is. Identify walks on $\mathbb{Z}$ with infinite binary sequences (say $0$ means moving ...
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### How might M.C. Escher have designed his patterns?

I realize this question isn't strictly mathematical, and if it doesn't fit with the content on this site then feel free (moderators/high-rep users) to close it. But when I thought up the question it ...
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### Parametrization of the boundary of the Mandelbrot set

Does anyone know how to parametrize the boundary of the Mandelbrot set? I am not a fractal-geometer or a dynamical systems person. I just have some idle curiosity about this question. The ...
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### local behavior of a finite Borel measure

Let $\mu$ be a finite Borel measure on $\mathbb{R}^n$. I am interested in how does $\mu(B(x,r))$ behave, where $B(x,r)$ is the open ball of radius $r$ centered at $x$. For instance, as far as I recall,...
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### Fractional Gaussian noise in higher dimensions

I'm having difficulty imagining (or explaining to myself) how the partial sums a two dimensional Gaussian noise can produce a surface. According to equation (20) of the paper, "On two-dimensional ...
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### Attractive Basins and Loops in Julia Sets

I recently learned about the Mandelbrot set for the first time from a presentation by some undergraduates in honor of Mandelbrot's death. The presentation was short and by non-experts so I left with ...